Fastest Mental Math Methods Compared: 14 Systems, Same 7 Problems

There is no single fastest method. With a tool, a trained soroban user is the fastest at general arithmetic. In your head, champion calculators use left to right and crosswise methods. For special numbers, Vedic shortcuts win. For any numbers at all, the general column methods (school, crosswise, lattice) never fail.

That is the short answer. This post is the long one. It runs fourteen calculation systems from two series on the same seven problems, counts the steps and the digits you have to hold in your head, and says plainly where a method cannot do a problem at all. The systems are: the Vedic sutras, Chinese counting rods and the suanpan, the Japanese soroban and its mental form anzan, the Trachtenberg speed system, Korean Chisanbop finger counting, Egyptian doubling, Russian peasant multiplication, line (stick) multiplication, Babylonian base 60 with reciprocal tables, Maya base 20 with bars and dots, lattice multiplication, Napier's bones, the slide rule, and the techniques used by competitors at the Mental Calculation World Cup. The school algorithm is the baseline.

A note on numbers. Every answer, running total, doubling column and base 60 or base 20 digit here was computed with Python; the script is in the walkthrough near the end. Counts not produced by the script are marked approximate. Historical claims come from MacTutor, the Yale Babylonian Collection, the Mental Calculation World Cup's published rules and results, and peer-reviewed studies. Speed is shown by counting, not asserted.

Key takeaways

• Every general multiplication method does the same single-digit products. For 386 x 524 they all use the same 9 products. They differ in the additions that follow, the size of the running sums and where intermediate results live.
• Special-case methods win big, but only on their cases. Nikhilam does 98 x 97 in 4 operations against 11 for the school method, and the "ends in 5" rule does 85² in 2 against 10. Neither helps with 73 x 46.
• The abacus moves the memory load out of your head. Soroban and suanpan need more moves than crosswise (24 against 17 for 386 x 524) but hold only one product in mind at a time.
• Trachtenberg's direct method is the lattice done in your head, and line multiplication is crosswise drawn as a picture. The first two keep the biggest running sum for 386 x 524 at 12, against 58 for crosswise; line multiplication makes you count 187 crossing points.
• Doubling methods (Egyptian, peasant) are slow by hand and fast in silicon. 386 x 524 takes 8 doublings and 2 additions of up to 6-digit numbers. The same shift and add idea runs in processors without a hardware multiplier.
• Some systems cannot do some problems. Chisanbop holds 0 to 99. Egyptian arithmetic writes 1/19 as a unit fraction and stops. Base 60 cannot give 1/19 or 1/37 exactly. No Maya source shows how they multiplied or divided.
• Measured records exist for the champions. At the 2024 Mental Calculation World Cup the winner scored 100 in a one-minute round of day-of-the-week dates and 25 in the 10-minute event for multiplying two 8-digit numbers, per the published results.
• Evidence on teaching is narrow. A three-year randomised controlled trial of mental abacus training with 204 children found better arithmetic and no change in basic cognitive abilities. We found no randomised trial of Vedic, Trachtenberg or Chisanbop instruction in an international peer-reviewed journal.

The fourteen systems at a glance

A word on the first row. The 16 sutras and 13 sub-sutras come from a book published in 1965, five years after its author died. Historians, including S. G. Dani and Kim Plofker in Mathematics in India (Princeton, 2009), note that these sutras do not appear in the Vedic texts. The methods can be fast and correct; they are twentieth-century methods. India's genuine early mathematics, from the Śulba Sūtras (शुल्बसूत्र) to Āryabhaṭa, Brahmagupta and the Kerala school, is a separate and much older history, covered in our Indian and Vedic mathematics post.

How we count steps, memory and carries

We use the same counting rules as our Vedic vs Chinese vs Japanese comparison, so the numbers in both posts line up.

• Core operations are the single-digit facts you use: each product such as 8 x 7, each addition or subtraction in a column, and each digit added onto an abacus rod. Placing a two-digit product on two rods counts as two rod additions.
• Row operations are whole doublings, halvings or additions in the doubling methods. We also give approximate digit counts, because doubling 67,072 is not one step for a human.
• Kept means the numbers you must hold before the answer appears. On an abacus the beads hold the running total, so the kept count in your head is close to zero. In anzan the image of the beads holds it.
• Largest running sum is the biggest number you have to add into at one time. It is a good proxy for working memory: adding 4 + 2 + 2 is easier than adding 12 + 16 + 30.
• Carries are the times a column or rod passes 9 and sends something left.
• Cannot means the method has no procedure for the problem in its own sources, not that a clever user could not improvise one.

Problem 1: 98 x 97 = 9,506

This is the problem the shortcut methods were built for. Both factors sit just below 100.

Vedic: Nikhilam with base 100

98   deficit 2
97   deficit 3
Left part:  98 − 3 = 95     (or 97 − 2)
Right part: 2 × 3  = 06     (two digits, because the base has two zeros)
Answer:     95 | 06 = 9,506

That is 4 operations, 2 numbers kept, no carries. It works because (100 − 2)(100 − 3) = 100 × (100 − 2 − 3) + 2 × 3.

Vedic: Urdhva-Tiryagbhyam (vertically and crosswise)

Left column:    9 × 9           = 81
Cross column:   9 × 7 + 8 × 9   = 63 + 72 = 135
Right column:   8 × 7           = 56
Combine from the right: 56 → write 6, carry 5; 135 + 5 = 140 → write 0, carry 14; 81 + 14 = 95
Answer: 9,506

That is 7 operations and 3 carries, with a largest running sum of 140.

Chinese counting board, suanpan, soroban and anzan

The board and both abacuses add each product in place, from the left.

Step   product      added at     running total on the rods
1      9 × 9 = 81   hundreds     8,100
2      9 × 7 = 63   tens         8,730
3      8 × 9 = 72   tens         9,450
4      8 × 7 = 56   units        9,506

That is 4 products and 8 rod additions, with 2 bead carries, and only one two-digit product in your head at a time. On the soroban each carry is a complement move: to add 7 to a rod showing 4, take away 3 and add 1 to the rod on the left. The full bead-by-bead version is in our Japanese mathematics post, and the rod layouts in our Chinese mathematics post.

Trachtenberg: the direct method with units and tens pairs

Trachtenberg's general multiplication builds the answer one digit at a time from the right. Each answer digit is the sum of the units digits of one set of products and the tens digits of the neighbouring set, plus the carry. Every number you add is a single digit.

Products: 9×9 = 81   9×7 = 63   8×9 = 72   8×7 = 56

Answer digit   units digits from     tens digits from    carry   sum   write
units          56 → 6                (none)              0       6     6
tens           63 → 3, 72 → 2        56 → 5              0       10    0, carry 1
hundreds       81 → 1                63 → 6, 72 → 7      1       15    5, carry 1
thousands      (none)                81 → 8              1       9     9
Answer: 9,506

That is 4 products and 6 single-digit additions, 10 operations in all, with 2 carries. The largest running sum is 15. The book also has quick rules for single multipliers: to multiply by 11, add each digit to its neighbour on the right; to multiply by 12, double each digit and add its neighbour. Those rules do not fit 97, so the direct method is the Trachtenberg route here.

Lattice and Napier's bones

The lattice writes each product in a cell split by a diagonal, tens above and units below, then adds along the diagonals. Our lattice, Napier's bones and slide rule post covers its history and the rods in detail.

          9      8
      +------+------+
      | 8 /  | 7 /  |
   9  |  / 1 |  / 2 |
      +------+------+
      | 6 /  | 5 /  |
   7  |  / 3 |  / 6 |
      +------+------+

Diagonal sums from the bottom right:
units:      6                 → 6
tens:       3 + 2 + 5 = 10    → 0, carry 1
hundreds:   1 + 7 + 6 + 1 = 15 → 5, carry 1
thousands:  8 + 1 = 9         → 9
Answer: 9,506

The diagonal sums (6, 10, 14, 8 before carries) are exactly the Trachtenberg column sums. The lattice is Trachtenberg's direct method on paper, or Trachtenberg is the lattice in your head: 10 operations, 2 carries. Napier's bones print the lattice cells on rods, so you read 98 x 9 and 98 x 7 off the bones instead of recalling the products, then add the two partial products.

Egyptian doubling

The full history and the 2/n table are in our Egyptian and Russian peasant multiplication post. Double 98 until the left column would pass 97, then tick the rows whose left numbers add to 97.

   1      98   ✓
   2     196
   4     392
   8     784
  16   1,568
  32   3,136   ✓
  64   6,272   ✓
97 = 64 + 32 + 1   →   6,272 + 3,136 + 98 = 9,506

That is 6 doublings and 2 additions, 8 row operations, about 27 digit steps. No multiplication table is needed at all, which is the method's great virtue.

Russian peasant multiplication

Halve the left column (dropping remainders), double the right, and add the right-hand numbers on rows where the left number is odd.

  97 (odd)      98    ✓
  48           196
  24           392
  12           784
   6         1,568
   3 (odd)   3,136    ✓
   1 (odd)   6,272    ✓
Sum: 98 + 3,136 + 6,272 = 9,506

That is 6 halvings, 6 doublings and 2 additions. The odd rows spell 97 in binary, 1100001, read from the bottom: the same rows the Egyptian scribe ticked. The peasant method simply finds the binary digits by halving instead of by subtraction.

Line (stick) multiplication

Draw 98 as 9 parallel lines and then 8 parallel lines, and cross them with 9 lines and 7 lines for 97. The crossing points fall into three diagonal zones, and you count each zone.

Zone (left to right)    crossings                 count
hundreds zone           9 lines × 9 lines         81
tens zone               9 × 7 + 8 × 9             135
units zone              8 lines × 7 lines         56
Carry exactly as in crosswise: 56 → 6 c5; 140 → 0 c14; 95 → 9,506

The zone counts are the Vedic crosswise column sums: this is crosswise multiplication drawn as a picture, at the cost of counting 272 crossing points on 33 lines. You will see it called "Japanese" or "Chinese" multiplication online; we found no historical source for either label, so treat both as folklore. It is a good way to show children why crosswise works, and slow for large digits.

Babylonian base 60

98 is 1,38 in base 60 (one sixty and 38), and 97 is 1,37. A scribe would multiply digit by digit using tables of multiples for each "head number".

         1 , 38
     ×   1 , 37
3600s:  1 × 1                = 1
60s:    1 × 37 + 38 × 1      = 75
units:  38 × 37              = 1,406
Normalise from the right:
1,406 = 23 × 60 + 26   → 26, carry 23
75 + 23 = 98 = 1 × 60 + 38 → 38, carry 1
1 + 1 = 2
Answer: 2,38,26   (2 × 3,600 + 38 × 60 + 26 = 9,506)

Four table look-ups, 3 additions, 2 carries. Base 60 makes the numbers short (three places for a five-digit decimal number) at the price of a much bigger multiplication table.

Maya base 20

98 is 4 twenties and 18, written vertically with the higher place on top: four dots over three bars and three dots. 97 is four dots over three bars and two dots. No surviving Maya source shows how they multiplied, so the layout below is a modern reconstruction in pure base 20.

98 = 4 | 18        97 = 4 | 17
400s:  4 × 4                  = 16
20s:   4 × 17 + 18 × 4        = 68 + 72 = 140
units: 18 × 17                = 306
Normalise from the bottom:
306 = 15 × 20 + 6     → 6,  carry 15
140 + 15 = 155        → 15 (155 = 7 × 20 + 15), carry 7
16 + 7 = 23           → 3,  carry 1
Answer: 1 | 3 | 15 | 6   (1 × 8,000 + 3 × 400 + 15 × 20 + 6 = 9,506)

In the calendar's Long Count, where the third place is 360 (18 x 20) instead of 400, the same number is 1.6.7.6 (7,200 + 2,160 + 140 + 6), so pure place value breaks at the third place.

Slide rule

Because log 98 + log 97 = log 9,506, the slide rule multiplies by adding lengths. You read about 9.51 and place the decimal point yourself: roughly 9,510. A standard rule gives about three significant figures, so the exact 9,506 is out of reach.

Chisanbop

Chisanbop shows 0 to 99 on two hands: right thumb 5, right fingers 1 each, left thumb 50, left fingers 10 each. It multiplies by repeated addition, so 98 x 97 means 97 additions of 98 with 95 hundreds tallied on paper. It works, and it is the slowest method here by a wide margin.

Champion techniques

A competitor sees "both near 100" and uses the identity (a − x)(a − y) = a(a − x − y) + xy: 95 hundreds plus 6 is 9,506. It is Nikhilam without the name.

School baseline

      9 8
    × 9 7
    -----
      6 8 6
    8 8 2 0
    -------
    9 5 0 6

4 products and 7 additions, 11 operations, 6 carries.

Verdict for problem 1: the near-base method (Vedic Nikhilam or the same identity used by champions) wins with 4 operations. The lattice and Trachtenberg take 10, the abacus 12 moves with almost nothing in memory, line multiplication 272 crossings to count, and Chisanbop 97 additions.

Problem 2: 47 x 53 = 2,491

The special feature here is symmetry: the factors sit 3 either side of 50.

Difference of squares (Vedic working base, champions, anyone)

47 × 53 = (50 − 3)(50 + 3) = 50² − 3² = 2,500 − 9 = 2,491

The Vedic book frames it as Nikhilam with a working base of 50 (about 6 operations); as algebra it is 3.

Crosswise, Trachtenberg and lattice

Vedic crosswise:    20 | 4×3 + 7×5 = 47 | 21    → 21: write 1 carry 2; 49: write 9 carry 4; 24 → 2,491
Trachtenberg pairs: units 1 (from 21); tens 5 + 2 (units of 35, 12) + 2 (tens of 21) = 9;
                    hundreds 0 (units of 20) + 3 + 1 (tens of 35, 12) = 4; thousands 2 (tens of 20)
Lattice diagonals:  1, 9, 4, 2   (no carries at all)

Line multiplication gives the same zone counts as crosswise (20, 47, 21) from 19 lines and 88 crossings. Vedic crosswise: 7 operations, 3 carries. Trachtenberg and the lattice: 4 products and 4 single-digit additions, 8 operations, no carries. This is a case where splitting products into units and tens removes every carry.

Abacus

Step   product      running total
1      4 × 5 = 20   2,000
2      4 × 3 = 12   2,120
3      7 × 5 = 35   2,470
4      7 × 3 = 21   2,491

4 products and 7 rod additions, no bead carries. The easiest of the four multiplications on a soroban.

Egyptian and peasant

Egyptian (double 47, 53 = 32 + 16 + 4 + 1):     Peasant (halve 47, double 53):
   1     47  ✓                                    47 odd     53  ✓
   2     94                                       23 odd    106  ✓
   4    188  ✓                                    11 odd    212  ✓
   8    376                                        5 odd    424  ✓
  16    752  ✓                                     2        848
  32  1,504  ✓                                     1 odd  1,696  ✓
47 + 188 + 752 + 1,504 = 2,491                   53 + 106 + 212 + 424 + 1,696 = 2,491

Egyptian: 5 doublings, 3 additions. Peasant: 5 halvings, 5 doublings, 4 additions. Which factor you decompose matters: 47 is 101111 in binary (five 1s), 53 is 110101 (four 1s).

Babylonian, Maya, Napier, slide rule and Chisanbop

• Base 60: 47 and 53 are single base 60 digits, so 47 x 53 is one look-up in a table of multiples: 41,31 (41 x 60 + 31 = 2,491). Here base 60 shines.
• Base 20: 2 | 7 times 2 | 13. Products 4, 26 + 14 = 40, 91. Normalising gives 6 | 4 | 11 (2,400 + 80 + 11 = 2,491). Again a reconstruction.
• Napier's bones: lay the 4 and 7 bones together and read rows 5 and 3: 235 and 141, then 2,350 + 141 = 2,491.
• Slide rule: about 2.49, so 2,490. Three figures.
• Chisanbop: 47 additions of 53 (or the other way round), with 24 hundreds tallied. Slow.

Verdict for problem 2: whoever spots the difference of squares wins in 3 operations. If you do not spot it, the lattice, Trachtenberg and the abacus all run carry-free.

Problem 3: 386 x 524 = 202,264

This is the fair test. Neither factor is near a power of 10, they are not symmetric about a round number, and nothing ends in 5. Only general methods apply.

Vedic crosswise

Column 1:  3 × 5                     = 15
Column 2:  3 × 2 + 8 × 5             = 46
Column 3:  3 × 4 + 8 × 2 + 6 × 5     = 12 + 16 + 30 = 58
Column 4:  8 × 4 + 6 × 2             = 44
Column 5:  6 × 4                     = 24
Combine: 24 → 4 c2; 46 → 6 c4; 62 → 2 c6; 52 → 2 c5; 20
Answer: 202,264

9 products and 8 additions, 17 operations, 5 carries. Fewest operations of any general method, but the middle column asks you to hold a running sum of three two-digit products, up to 58, and the carries reach 6.

Trachtenberg direct method and the lattice

Products:  3×5=15  3×2=06  3×4=12  8×5=40  8×2=16  8×4=32  6×5=30  6×2=12  6×4=24

Lattice (tens / units in each cell):
            5        2        4
       +--------+--------+--------+
   3   |  1 / 5 |  0 / 6 |  1 / 2 |
       +--------+--------+--------+
   8   |  4 / 0 |  1 / 6 |  3 / 2 |
       +--------+--------+--------+
   6   |  3 / 0 |  1 / 2 |  2 / 4 |
       +--------+--------+--------+

Answer digit    single digits added                    sum   write
units           4                                      4     4
tens            2 + 2 + 2                              6     6
hundreds        2 + 6 + 0 + 1 + 3                      12    2, carry 1
thousands       6 + 0 + 1 + 1 + 3, + carry 1           12    2, carry 1
ten thousands   5 + 4 + 0, + carry 1                   10    0, carry 1
hundred thous.  1, + carry 1                           2     2
Answer: 202,264

9 products and 15 single-digit additions, 24 operations, 3 carries, and the largest running sum is 12. Same operation count as the abacus, with almost the same memory profile. On paper this is the lattice. In your head it is Trachtenberg's direct method.

Line multiplication

Five zones with 15, 46, 58, 44 and 24 crossings, the crosswise column sums, from 28 lines and 187 crossing points. Correct, and far too much counting to be fast; the middle zone alone has 58 points.

Napier's bones

Lay the bones for 3, 8 and 6 side by side. Each row of the bones is a row of the lattice, with the diagonals already drawn.

Row 5:  1/5  4/0  3/0   → diagonals from the right: 0; 3 + 0 = 3; 4 + 5 = 9; 1   → 1,930
Row 2:  0/6  1/6  1/2   → 2; 1 + 6 = 7; 1 + 6 = 7; 0                             → 772
Row 4:  1/2  3/2  2/4   → 4; 2 + 2 = 4; 3 + 2 = 5; 1                             → 1,544
Sum:    193,000 + 7,720 + 1,544 = 202,264

No products recalled at all; about 6 diagonal additions and 9 column additions. Napier's bones made the times table unnecessary, which mattered when most clerks had not memorised it.

Abacus: suanpan or soroban

Step   product      running total
1      3 × 5 = 15   150,000
2      3 × 2 = 6    156,000
3      3 × 4 = 12   157,200
4      8 × 5 = 40   197,200
5      8 × 2 = 16   198,800
6      8 × 4 = 32   199,120
7      6 × 5 = 30   202,120
8      6 × 2 = 12   202,240
9      6 × 4 = 24   202,264

9 products and 15 rod additions, 3 bead carries. Nothing in your head except the product you just recalled.

Egyptian and peasant

Choose the factor with fewer binary 1s as the one to decompose. 386 is 110000010 in binary, three 1s.

Egyptian (double 524; 386 = 256 + 128 + 2):
    1        524
    2      1,048   ✓
    4      2,096
    8      4,192
   16      8,384
   32     16,768
   64     33,536
  128     67,072   ✓
  256    134,144   ✓
1,048 + 67,072 + 134,144 = 202,264

8 doublings and 2 additions, 10 row operations, but the doublings run to six digits: about 45 digit steps. The peasant layout halves 386 instead (386, 193, 96, 48, 24, 12, 6, 3, 1) and adds the rows beside 193, 3 and 1: 1,048 + 67,072 + 134,144. Same sum, same cost.

Babylonian base 60

386 is 6,26 and 524 is 8,44. The product is 56,11,4 (56 x 3,600 + 11 x 60 + 4 = 202,264). The four digit products (6 x 8, 6 x 44, 26 x 8, 26 x 44 = 1,144) each come from a table of multiples: four look-ups instead of nine products, at the cost of carrying tables for every head number.

Maya base 20

In pure base 20, 386 is 19 | 6 and 524 is 1 | 6 | 4. The product is 1 | 5 | 5 | 13 | 4. Six digit products of numbers up to 19 and a long normalisation: possible as a reconstruction, with no evidence the Maya did it this way.

Slide rule

log 386 + log 524 = 2.5866 + 2.7193 = 5.3059, which is the log of 202,264. On the rule you read about 2.02 and place the point: 202,000. The error is 0.13 percent, fine for an engineer's estimate and wrong for an invoice.

Champion techniques

Calculators at the World Cup multiply from the left so the first digits they say are final. One route:

386 × 500 = 193,000
386 × 24  = 386 × 20 + 386 × 4 = 7,720 + 1,544 = 9,264
193,000 + 9,264 = 202,264

Same nine digit facts; the answer comes out from the left.

Chisanbop

Cannot, in any practical sense. Neither factor fits on two hands (0 to 99), so you would split both factors, run hundreds of repeated additions and keep most of the bookkeeping on paper.

School baseline

9 products and 17 additions, 26 operations, 11 carries, 3 partial product rows.

Verdict for problem 3: Vedic crosswise has the fewest operations (17) and the heaviest memory load. The lattice, Trachtenberg and the abacus take 24 with tiny running sums. Napier's bones remove the products; the slide rule wins if three figures are enough.

Problem 4: 85² = 7,225

Vedic Ekadhikena Purvena and champion squaring

Vedic:     8 × (8 + 1) = 72, append 25          → 7,225
Champion:  85² = (85 + 5)(85 − 5) + 5² = 90 × 80 + 25 = 7,200 + 25 = 7,225

Vedic: 2 operations. The champion identity, a² = (a + b)(a − b) + b², is more general: it works for 86² too (90 x 82 + 16 = 7,396), which the "ends in 5" rule does not.

General methods

• Abacus: products 64, 40, 40, 25 give running totals 6,400, 6,800, 7,200, 7,225. 4 products, 6 rod additions, 1 bead carry.
• Trachtenberg and lattice: column sums 5, 2, 12 and 6, then 5; 2; 12 → 2 carry 1; 6 + 1 = 7. 4 products and 5 additions, 9 operations, 1 carry.
• School: 425 + 6,800 = 7,225; 10 operations, 5 carries.
• Line multiplication: zones of 64, 80 and 25 crossings (169 points from 26 lines), then 25 → 5 c2; 82 → 2 c8; 72 → 7,225.
• Egyptian: 85 = 64 + 16 + 4 + 1, so 6 doublings of 85 (to 5,440) and 3 additions: 85 + 340 + 1,360 + 5,440 = 7,225.
• Peasant: halving 85 gives 85, 42, 21, 10, 5, 2, 1, with four odd rows: 85 + 340 + 1,360 + 5,440.
• Base 60: 85 is 1,25; the square is 2,0,25. That middle 0 is a trap: Old Babylonian scribes wrote no zero sign, so 2,0,25 and 2,25 could look alike, and context had to decide.
• Base 20: 85 is 4 | 5 and 7,225 is 18 | 1 | 5. The Maya shell zero could mark an empty place.
• Slide rule: read about 7.22 or 7.23 on the A scale, placed as roughly 7,220 to 7,230.
• Chisanbop: 85 additions of 85.

Verdict for problem 4: the special-case rule wins (2 operations), and the near-round identity is the champion's general version of it.

Problem 5: √7,744 = 88

Roots separate the systems more sharply than any multiplication.

Inspection (Vedic and champions)

Last digit 4 → root ends in 2 or 8
First pair 77 lies between 64 (8²) and 81 (9²) → tens digit 8
82 or 88? 8 × 9 = 72 and 77 is above 72 → the larger, 88
Check: 88² = 7,744

3 operations, but only for numbers you already know are perfect squares. Champions use the same reasoning for cube roots of perfect cubes: 704,969 ends in 9, so the cube root ends in 9; 704 lies between 8³ = 512 and 9³ = 729, so the tens digit is 8; the root is 89, and 89³ = 704,969.

Digit-by-digit methods (Vedic duplex, Chinese kaifang, soroban, school)

First digit: largest square not above 77 is 64 → 8, remainder 13
Bring down 44 → 1,344; trial divisor 2 × 80 = 160
(160 + d) × d should not exceed 1,344 → d = 8, 168 × 8 = 1,344, remainder 0
Root: 88

About 4 to 5 operations. This procedure appears on the Chinese counting board in the Nine Chapters, on the soroban as part of dan level grading, and on paper as the school long-division root.

Babylonian: the averaging method

The Babylonian method, later written down by Heron, improves a guess by averaging it with the number divided by the guess. Start at 90:

Guess 90:            (90 + 7,744 ÷ 90) ÷ 2 = (90 + 86.044...) ÷ 2 = 88.0222...
Guess 88.0222...:    next average = 88.0000028...
Guess 88.0000028...: next average = 88.0000000000000...

Two rounds give 88 to five decimal places; the number of correct digits roughly doubles each round. The famous tablet YBC 7289 in the Yale Babylonian Collection, dated to about 1800 to 1600 BCE, writes √2 as 1;24,51,10 in base 60. That is 1.41421296..., against the true 1.41421356..., an error of about 0.0000006, within one unit in the sixth decimal place. In base 60, 7,744 is 2,9,4 and its root 88 is 1,28.

Another Old Babylonian tablet is the most argued-over in the subject. Plimpton 322, in Columbia University's collection and dated by MacTutor to between 1800 and 1650 BCE, lists numbers that fit Pythagorean triples. Eleanor Robson (American Mathematical Monthly, 2002) reads it as a teacher's aid generated from pairs of reciprocals, in line with how scribes worked. Daniel Mansfield and Norman Wildberger (Historia Mathematica, 2017) argue it is an exact trigonometric table in base 60. The debate is open, and the reciprocal-pair reading is the one that ties it to the fast division method above.

Trachtenberg

The 1960 book has a chapter on squares and square roots; we leave its exact root layout to the Trachtenberg deep dive listed below.

Napier's bones

MacTutor notes that Napier's set was used for square roots and cube roots as well as multiplication and division. The extra rods give the trial products (160 + d) × d of the digit-by-digit method above, so you read 1,344 off the rods instead of computing it.

Slide rule

Set 7,744 on the A scale and read the D scale: about 88.0. Here the slide rule is exact to three figures, which for this number is the whole answer.

Methods that cannot

• Egyptian: the surviving papyri include problems that need square roots, but no general extraction procedure. A scribe could test candidates by doubling: 88 x 88 = 7,744 checks, but finding 88 is guesswork.
• Maya: no surviving source shows root extraction.
• Chisanbop: no root procedure. You could test 88 x 88 by 88 additions of 88.
• Lattice and line multiplication: multiplication layouts only; they can check 88² but not find 88.
• Peasant multiplication: the same; it multiplies, it does not invert.

Verdict for problem 5: for a known perfect square, inspection in 3 steps. For an unknown number, the digit-by-digit procedure (Chinese, soroban, school) or the Babylonian average, which converges fastest of all when you can divide accurately.

Problem 6: 7,429 ÷ 37 = 200 remainder 29

Vedic flag method (Dhvajanka) and Trachtenberg division

Split 37 into a main digit 3 and a flag 7, and divide by 3 only, subtracting flag corrections as you go.

Dividend 7 4 2 | 9
Digit 7:  7 ÷ 3 = 2 remainder 1                            → quotient 2
Digit 4:  14 − 7 × 2 = 0; 0 ÷ 3 = 0                        → quotient 0
Digit 2:  2 − 7 × 0 = 2; 2 ÷ 3 = 0 remainder 2             → quotient 0
Remainder column: 29 − 7 × 0 = 29                          → remainder 29
Answer: 200 remainder 29

About 7 operations. Trachtenberg's division works in the same family: divide each partial dividend by the leading digit of the divisor and subtract the units and tens pairs formed with the rest of the divisor.

Counting board and soroban

On the Chinese board the divisor 37 sits under 74, the quotient digit 2 goes on top and 74 comes off directly, leaving 29. On the soroban, shōjohō (商除法) subtracts 2 x 3 and 2 x 7 straight from the rods. About 5 operations each, with the quotient and remainder both visible on the board at the end.

Egyptian division by doubling

Double the divisor and pick the rows that fit.

   1      37
   2      74
   4     148
   8     296   ✓
  16     592
  32   1,184
  64   2,368   ✓
 128   4,736   ✓
7,429 − 4,736 = 2,693;  2,693 − 2,368 = 325;  325 − 296 = 29
Quotient 128 + 64 + 8 = 200, remainder 29

7 doublings and 3 subtractions. MacTutor's account of the Rhind papyrus shows exactly this layout for 1,495 ÷ 65. It is slow to set up and very reliable, because every step is a doubling or a subtraction.

Babylonian: the reciprocal problem

Babylonian division meant multiplying by a reciprocal from a table, and the standard tables list only "regular" numbers, those whose prime factors are 2, 3 and 5, because only those have finite base 60 reciprocals. 37 is prime and not regular. Its reciprocal in base 60 never ends: 0;1,37,17,50,16,12 and on. A scribe would use an approximate reciprocal, giving roughly 200;47 (200.78) and no clean remainder. Fast for divisors like 2, 3, 4, 5, 6, 8, 9, 12 and 15; awkward for 7, 11, 13, 19 or 37.

Champion trick: 37 x 27 = 999

7,429 ÷ 37 = 7,429 × 27 ÷ 999
7,429 × 27 = 200,583
200,583 = 200 × 999 + 783, and 783 = 29 × 27
So 7,429 ÷ 37 = 200 remainder 29, or 200.783783...

Dividing by 999 is easy because it just repeats a three-digit block, but multiplying 7,429 by 27 first costs about 15 operations, so on this problem the trick is elegant rather than fast. It is specific to divisors of 999 (27, 37, 111 and so on), and it shines when you want the repeating decimal: 200.783783...

Others

• Napier's bones: lay out the 3 and 7 bones to read every multiple of 37 at once (37, 74, 111, 148, ...), then run long division picking rows. About 5 operations, no products recalled.
• Slide rule: about 201. You get 200.8 at best and no remainder.
• Lattice and peasant: multiplication layouts. They can check 200 x 37 + 29, not perform the division.
• Maya: no surviving division procedure.
• Chisanbop: repeated subtraction of 37 two hundred times. Cannot in practice.
• School: about 8 operations.

Verdict for problem 6: the board and soroban are quickest (about 5), the flag method keeps the divisor to one digit, and Egyptian doubling is slowest but nearly error-proof. Base 60 struggles because 37 is not regular.

Problem 7: 1/19 as a decimal = 0.052631578947368421, repeating

The block repeats every 18 digits.

Vedic Ekadhikena Purvena and champions

For a denominator ending in 9, "one more than the one before" is 2, so you divide by 2 and feed each remainder forward as a prefix.

1 ÷ 2 = 0 r 1 → 0      10 ÷ 2 = 5 r 0 → 5      5 ÷ 2 = 2 r 1 → 2
12 ÷ 2 = 6 r 0 → 6     6 ÷ 2 = 3 r 0 → 3       3 ÷ 2 = 1 r 1 → 1
11 ÷ 2 = 5 r 1 → 5     15 ÷ 2 = 7 r 1 → 7      17 ÷ 2 = 8 r 1 → 8
First 9 digits: 052631578
Next 9 = nines complements: 947368421     (052631578 + 947368421 = 999,999,999)
1/19 = 0.052631578947368421 ...

9 halvings and 9 subtractions from 9, every number at most 19. Champions use the same pattern, often from the right by doubling: 1, 2, 4, 8, 16 (write 6, carry 1), 13 (write 3, carry 1), and so on.

Long division methods

Counting board, soroban and school long division all run 18 trial divisions by 19, each with a product and a subtraction: about 54 operations. The board and soroban are faster per step but not fewer steps.

Egyptian: unit fractions

An Egyptian scribe would not convert 1/19 at all. It is already a unit fraction, the form Egyptian arithmetic used for every fraction. Its double comes from the Rhind papyrus 2/n table: 2/19 = 1/12 + 1/76 + 1/114. So the Egyptian answer to "1/19" is "1/19", and that is the honest comparison: the system had no decimals to produce.

Babylonian base 60

19 is not regular, so 1/19 has no finite base 60 expansion either. It begins 0;3,9,28,25,15,47,22,6 and repeats with period 18 in base 60 as well. A scribe would use an approximation. Base 60 handles 1/2, 1/3, 1/4, 1/5, 1/6, 1/8, 1/9, 1/10, 1/12 and 1/15 exactly, which decimals cannot all do; 1/3 in base 60 is simply 0;20.

Methods that cannot

• Maya: no fractions in the surviving sources.
• Chisanbop: no fraction or decimal procedure.
• Lattice, line and peasant multiplication: not division tools. Napier's bones can support the long division, as for problem 6.
• Slide rule: about 0.0526. Three figures, no repeating block.
• Trachtenberg: the division method applies, at the same 18 steps as long division.

Verdict for problem 7: the Vedic and champion halving or doubling pattern wins outright, and its nines complement property is a built-in check.

The metrics table: every method, every problem

Core operation counts follow the rules above. "Cannot" means no procedure in the method's own sources. Counts marked "rows" are whole-number doublings, halvings or additions.

And the per-method view, for the questions that do not depend on one problem:

Three patterns stand out.

First, the shortcut methods win by a factor of 2 to 5 exactly where they apply, and they apply to a minority of numbers.

Second, on general multiplication every method does the same digit products. The differences are in memory load, not mathematical power: the lattice, Trachtenberg and the abacus trade more operations for smaller numbers to hold.

Third, "cannot" is not a failure of the users. Egyptian arithmetic did not need decimals, the surviving Maya sources are calendrical and astronomical, and Chisanbop was designed for early-grade addition and subtraction, which it does well.

What measured records and research show

"Fast" claims are easy to make and hard to check. Here is what is measured.

Mental Calculation World Cup

The Mental Calculation World Cup has run since 2004, when Robert Fountain of Great Britain won the first event in Annaberg-Buchholz, Germany, according to its organisers. Its 2024 rules fix four categories (weekdays for dates from 1600 to 2100, adding ten 10-digit numbers, multiplying two 8-digit numbers, and square roots of 6-digit numbers to eight significant digits) plus surprise problems. Writing down intermediate results is not allowed. In the addition, multiplication and root categories, each correct answer scores a point and each wrong answer loses one.

At the 2024 event in Paderborn, 35 calculators from 17 countries took part. The published results show the overall winner, Aaryan Nitin Shukla of India, with:

For comparison, the 2004 results list 33 correct calendar dates, in two one-minute series, as a world record at the time, and Alberto Coto of Spain solving 10 of 10 ten-number additions in 5 minutes 50 seconds. The formats changed, so this is not a clean like-for-like trend.

Memoriad, a separate organisation that runs memory and mental calculation championships, lists events such as square roots of 6, 8 and 10 digit numbers and dividing a 10-digit number by a 5-digit one. Its 2024 international edition was held in Dubai on 7 to 9 November 2024.

The calendar example, worked

The World Cup rules use 8 December 1721 as their sample date, with the answer Monday. Here it is by John Conway's Doomsday rule.

Century anchor for 1700s: Sunday
Year 21: 21 ÷ 12 = 1 remainder 9; 9 ÷ 4 = 2 (whole part); 1 + 9 + 2 = 12; 12 mod 7 = 5
Doomsday 1721 = Sunday + 5 = Friday
December's doomsday is 12/12, so 12 December 1721 was a Friday
8 December is 4 days earlier: Monday

Peer-reviewed studies, with their limits

• Mental abacus, randomised trial. Barner and colleagues (Child Development, 2016) followed 204 elementary school children for three years, starting at ages 5 to 7. Mental abacus students outperformed controls on arithmetic. The training did not change basic cognitive abilities such as working memory; children's spatial working memory at the start predicted how much they gained. One trial, one country, arithmetic outcomes only.
• Mental abacus, mechanism. Frank and Barner (Journal of Experimental Psychology: General, 2012) found that users store the image column by column and resist verbal interference, which fits a visual process. Earlier studies include Hatano, Miyake and Binks (Cognition, 1977) and Stigler (Cognitive Psychology, 1984).
• Vedic, Trachtenberg and Chisanbop. We searched for randomised or controlled trials of each in international peer-reviewed journals and found none we could cite. Chisanbop had a wave of US classroom adoption after 1977, and contemporary press reports describe mixed results; we could not retrieve those reports to verify them, so we draw no conclusion from them.

None of this supports claims about intelligence, brain development or exam results, and we make none.

Which method is fastest for what

With a tool: the soroban (or suanpan)

For general arithmetic with a tool in hand, the abacus is the fastest system with the strongest evidence. High move counts, lowest memory load, and the moves become motor habits. Napier's bones are the historical runner-up, and the slide rule wins if three significant figures are enough.

In your head: champion techniques built on crosswise and left to right

For mental work on arbitrary numbers, the champions' toolkit wins: left to right multiplication so the answer comes out in speaking order, crosswise columns (the same as Vedic Urdhva) for compact products, and anzan for those trained on the soroban. Trachtenberg's units and tens pairs are a good choice for learners because they keep every running sum small.

For special cases: Vedic sutras and near-round identities

Near a base (98 x 97), symmetric about a round number (47 x 53), ending in 5 (85²), known perfect squares (7,744), denominators ending in 9 (1/19) and divisors of 999 (37): these are where the special-case methods cut work by half or more. The skill is spotting the case in under a second.

In general, on paper: the column methods

The school algorithm, the lattice and Egyptian doubling never fail on whole-number multiplication. The lattice is the easiest to check line by line. Egyptian doubling needs no multiplication table at all.

A practice plan: 12 weeks, 20 minutes a day

This plan takes the best of each system. It assumes you know the times table.

The habit that matters most is the week 12 one: look first, then choose.

Walkthrough: reproduce every number in Python

Save this as compare.py and run python3 compare.py. It uses only the standard library and prints every answer, doubling column, base 60 and base 20 digit, lattice diagonal and operation count used above.

from fractions import Fraction
import math, datetime

def ut_pairs(a, b):
    """Trachtenberg units and tens pairs, which are the lattice diagonals."""
    A = [int(d) for d in str(a)][::-1]
    B = [int(d) for d in str(b)][::-1]
    carry, digits, adds, carries, biggest = 0, [], 0, 0, 0
    for k in range(len(A) + len(B)):
        terms = [(A[i] * B[k - i]) % 10 for i in range(len(A)) if 0 <= k - i < len(B)]
        terms += [(A[i] * B[k - 1 - i]) // 10 for i in range(len(A)) if 0 <= k - 1 - i < len(B)]
        if carry:
            terms.append(carry)
        adds += max(len(terms) - 1, 0)
        s = sum(terms)
        biggest = max(biggest, s)
        digits.append(s % 10)
        carry = s // 10
        carries += carry > 0
    value = int("".join(map(str, digits[::-1])))
    return value, len(A) * len(B), adds, carries, biggest

def doubling(multiplier, other):
    """Egyptian doubling: rows, and the rows whose left numbers sum to the multiplier."""
    rows, p, v = [], 1, other
    while p <= multiplier:
        rows.append((p, v))
        p, v = p * 2, v * 2
    picked, rest = [], multiplier
    for p, v in reversed(rows):
        if p <= rest:
            picked.append(v)
            rest -= p
    return rows, picked, sum(picked)

def peasant(a, b):
    rows = []
    while a >= 1:
        rows.append((a, b))
        a, b = a // 2, b * 2
    return rows, sum(b for a, b in rows if a % 2)

def digits_in_base(n, base):
    out = []
    while n:
        out.append(n % base)
        n //= base
    return out[::-1]

def long_count(n):
    out = []
    for b in (20, 18, 20, 20, 20):
        out.append(n % b)
        n //= b
        if n == 0:
            break
    return out[::-1]

for a, b in [(98, 97), (47, 53), (386, 524), (85, 85)]:
    print(a, "x", b, "=", a * b, "pairs/lattice:", ut_pairs(a, b))
    multiplier = {(98, 97): 97, (47, 53): 53, (386, 524): 386, (85, 85): 85}[(a, b)]
    print("  egyptian:", doubling(multiplier, a * b // multiplier)[1:])
    print("  peasant:", peasant(a, b)[1])
    print("  base 60:", digits_in_base(a * b, 60), "base 20:", digits_in_base(a * b, 20),
          "long count:", long_count(a * b))

print("sqrt 7744:", math.isqrt(7744), "heron from 90:", end=" ")
x = 90.0
for _ in range(3):
    x = (x + 7744 / x) / 2
    print(round(x, 10), end=" ")
print()
ybc = 1 + Fraction(24, 60) + Fraction(51, 3600) + Fraction(10, 216000)
print("YBC 7289:", float(ybc), "error", float(ybc) - math.sqrt(2))

print("7429 / 37:", divmod(7429, 37), "via 27/999:", divmod(7429 * 27, 999))
rows, picked, total = doubling(200, 37)
print("egyptian division rows:", rows)

r, digits = 1, []
for _ in range(18):
    r *= 10
    digits.append(r // 19)
    r %= 19
print("1/19 = 0." + "".join(map(str, digits)))
print("2/19 =", Fraction(1, 12) + Fraction(1, 76) + Fraction(1, 114))
x, sexa = Fraction(1, 19), []
for _ in range(8):
    x *= 60
    sexa.append(int(x))
    x -= int(x)
print("1/19 in base 60: 0;" + ",".join(map(str, sexa)))

for d in [datetime.date(1721, 12, 8), datetime.date(2026, 10, 25)]:
    print(d, d.strftime("%A"))
print("89 cubed:", 89 ** 3)

Common mistakes

• Calling the Vedic sutras ancient. They come from a book published in 1965. Use them, and credit India's genuine classical mathematics separately.
• Using a near-base method far from the base. For 61 x 58 with base 100 the deficits are 39 and 42, and 39 x 42 is harder than the original problem. Pick base 60, or use a general method.
• Comparing operation counts without memory load. Vedic crosswise uses fewer operations than the abacus for 386 x 524 (17 against 24), but asks you to hold a running sum of 58. Count both.
• Treating the slide rule as exact. It gives three significant figures. 386 x 524 reads as 202,000, not 202,264.
• Forgetting that base 60 needs regular divisors. 1/37 and 1/19 never terminate in base 60. Reciprocal tables skip them.
• Treating casting out nines as proof. It misses swapped digits: 9,056 has the same digit sum as 9,506. Add the elevens check or multiply back.
• Expecting Chisanbop to scale. It is an addition and subtraction tool for 0 to 99. It is not a general calculator.

FAQ

What is the fastest mental math method?

For arbitrary numbers, measured competition results favour trained mental calculators using left to right and crosswise methods, and mental abacus (anzan) users. For special numbers, Vedic shortcuts take the fewest steps. No single method is fastest on every problem.

Is Vedic maths faster than abacus?

On special cases such as 98 x 97 or 85², yes: 4 and 2 operations against 12 and 10 abacus moves. On general products, a trained soroban user is faster, and the mental abacus has the stronger evidence, including a three-year randomised controlled trial.

Is the Trachtenberg system still worth learning?

Yes, as a way to multiply with only single-digit additions. Its direct method is the lattice done in your head, and its running sums stay small (12 for 386 x 524). We found no controlled trial of it, so treat claims about learning benefits with care.

How high can you count with Chisanbop?

To 99. The right hand shows 0 to 9 (thumb 5, fingers 1) and the left shows tens (thumb 50, fingers 10). Above 99 you tally on paper.

Why does Russian peasant multiplication work?

Halving writes the left number in binary, and each odd row is a 1 bit, so the odd rows add the other factor times each power of 2 in the first. Processors without a hardware multiplier use the same shift and add loop.

Why did the Babylonians use base 60?

Nobody knows for certain. Base 60 has many divisors (2, 3, 4, 5, 6, 10, 12, 15, 20, 30), so many reciprocals are short and division by multiplying with a reciprocal is fast. It survives in our 60 minutes and 360 degrees.

Did the Maya have a zero?

Yes, a shell sign in their positional numbers, an independent invention of zero. The calendar's Long Count used 360 instead of 400 in the third place.

What is the Doomsday rule?

John Conway's method for the day of the week. Each year has a "doomsday" weekday shared by dates such as 4/4, 6/6, 8/8, 10/10 and 12/12; you find it from a century anchor and count to the date. The 2024 World Cup winner scored 100 dates in one minute.

Deep dives

The four mathematics traditions posts are live:

• Indian and Vedic Mathematics Explained: the 16 Sutras, Śulba Sūtras, Zero, Āryabhaṭa, Brahmagupta and the Kerala School
• Chinese Mathematics Explained: Counting Rods, the Nine Chapters, Fangcheng, the Remainder Theorem and Horner before Horner
• Japanese Mathematics Explained: the Soroban, Anzan, Wasan, Seki Takakazu and Sangaku Temple Geometry
• Vedic vs Chinese vs Japanese Math: Same Problems, Three Systems

The fast calculation methods series, one post per method:

• Trachtenberg Speed System Explained: Multiplication Rules for 11 to 12 and Any Number, with Worked Examples
• Chisanbop Korean Finger Math Explained: Counting to 99 on Two Hands, with Worked Examples
• Egyptian and Russian Peasant Multiplication Explained: Doubling, Halving, Binary and Egyptian Fractions
• Babylonian Mathematics Explained: Base 60, Reciprocal Tables, YBC 7289 and Plimpton 322
• Maya Mathematics Explained: Base 20, Bar and Dot Numerals, Zero and Long Count Calendar Arithmetic
• Lattice Multiplication, Napier's Bones and the Slide Rule Explained: Mechanical Shortcuts before Calculators
• Mental Calculation Techniques of World Champions: Left to Right, Cross Multiplication, Squaring and Calendar Tricks
• Line Multiplication Explained: the Stick Method that Counts Crossing Points, Japanese, Chinese or Neither?

Checksums, doubling and checkable steps: an honest bridge

Safeguard does not do mathematics, and nothing in this post was computed by a Safeguard model; every number came from the plain Python script above. The parallel to our work is in habits. Casting out nines is a checksum, the same idea as the hash that confirms a downloaded package is the one that was published. The peasant method's halve-and-double is the binary arithmetic underneath every program. And the methods that win here all keep each intermediate step visible and checkable, which is exactly what a software supply chain needs from source to build to release. If you build a drill app, a classroom quiz site or a notebook around these methods, it runs on open source dependencies like any other software, and those need the same checking and timely fixes.

More in this series

Other posts in our fast calculation methods series:

• Babylonian Mathematics Explained: Base 60, YBC 7289 and Plimpton 322
• Chisanbop Explained: Korean Finger Math to 99, With Worked Examples
• Egyptian and Russian Peasant Multiplication: Doubling and Binary
• Lattice Multiplication, Napier's Bones and the Slide Rule Explained
• Line Multiplication: The Stick Method, Japanese, Chinese or Neither?
• Maya Mathematics Explained: Base 20, Bar and Dot Numerals and Zero
• Mental Calculation Techniques of World Champions: Left to Right, Criss-Cross, Squares and Calendars
• Trachtenberg Speed System Explained: Rules for 11, 12 and Any Number

Fix it without waiting for upstream

Whatever the vulnerability, most of the risk sits in two gaps. The first is the time between a fix being published and that fix actually running in your production builds. The second, and worse, is the time before any fix exists at all, while a maintainer works out a patch and a release schedule. Waiting through both gaps is a habit, not a requirement. Safeguard gives teams three ways to close them.

1. Autonomous remediation: pull requests, not tickets

When a fixed version exists, Safeguard's Griffin models open a tested, policy gated pull request that moves the affected dependency to the safe version. You can run it on one repository or in bulk across every project that carries the vulnerable version. Remediation covers eight categories, including dependencies, licenses, security posture and findings. Reachability analysis ranks the vulnerable code paths your application actually calls first, so the first pull requests you merge are the ones that matter most. Autonomous remediation ships switched off, with every category set to Manual, so your team decides how much runs on its own and what still needs a human approval.

2. Gold Open Source: the safe version in seconds, free

gold.safeguard.sh is free and needs no login. Search any CVE or package to see affected and fixed versions, CVSS, EPSS, CISA KEV status and a recommended safe version, or use the "Am I affected?" checker. The same data is available through a free read API, README status badges and RSS threat feeds. The safeguard-sh/gold-scan-action GitHub Action can fail a build on a known exploited vulnerability, on any known vulnerability, or never, so a version you have just removed cannot quietly come back in next week's merge.

3. Gold Registry: start from components that are already fixed

registry.safeguard.sh provides hardened, drop in replacements for upstream packages and container images. Every Gold artifact is held continuously to zero exploitable critical or high CVEs, zero malware and a completed license check, with SLSA L3 or higher provenance, a Sigstore signature and a public evidence page you can hand to an auditor. An artifact loses its Gold badge if it regresses. Teams can point their own package directory at the registry so hardened equivalents resolve without changing how developers install anything.

When upstream has no fix yet, or will not ship one, Safeguard first works to get the fix accepted upstream. If the issue is severe or upstream will not move, Safeguard builds a fork, fixes the vulnerability there and publishes a hardened Gold build, so you are not stuck on a maintainer's release schedule. You can also request a hardened, compatibility tested build of a specific package. This fork and patch flow is in early access today, so ask which packages it covers before you plan around it.

One honest limit: when the vulnerable code lives in a vendor appliance, a hosted service or an operating system runtime you do not build yourself, the fix still has to come from that vendor or distribution. The steps above apply to the open source dependencies and container images you build and ship, which is where most teams carry the largest and least tracked share of their exposure.

See how Safeguard would help

None of these demos involve arithmetic. They show the software side described in the bridge above.

The first demo is a real Safeguard CLI session. It installs the CLI, opens the interactive terminal and asks it to fix a lodash CVE in a repository. It reads the finding and the advisory, asks before any command that changes something, edits package.json to lodash 4.17.21, runs npm install and npm test with approval, prints a summary and pushes nothing. The same flow fits a web app for soroban or mental math drills: scan the repository, then let the assistant propose the dependency upgrade under your approval.

Click through it at your own pace, or open it full screen.

The second demo sets up the Safeguard runner on your own machine, approves it with its device code and then scans from the terminal. That suits teachers and hobbyists who keep scripts like the walkthrough above on their own laptops and want the scan to run where the code lives.

Click through it at your own pace, or open it full screen.

The third demo connects Claude to the Safeguard MCP server with OAuth and lets you choose which tools Claude can use. If you already work with Claude while building an education app, it brings dependency findings into the same conversation, limited to the tools you allow.

Click through it at your own pace, or open it full screen.

To try it on your own repositories, download Safeguard and follow the CLI documentation.

Quick check

Sources

• Mental Calculation World Cup, overview of all events (recordholders.org)
• Mental Calculation World Cup 2024, general information
• Mental Calculation World Cup 2024, results
• Mental Calculation World Cup 2004, results
• Rules for the Mental Calculation World Cup 2024 (PDF)
• Memoriad, World Memory, Mental Calculation and Speed Reading Organization
• MacTutor History of Mathematics, Egyptian papyri
• MacTutor History of Mathematics, An overview of Babylonian mathematics
• MacTutor History of Mathematics, Babylonian numerals
• MacTutor History of Mathematics, Pythagoras's theorem in Babylonian mathematics (YBC 7289 and Plimpton 322)
• Yale Babylonian Collection tablet YBC 7289, photographs and commentary by Bill Casselman
• MacTutor History of Mathematics, Mayan mathematics
• MacTutor History of Mathematics, John Napier
• MacTutor History of Mathematics, Leonardo Pisano (Fibonacci)
• MacTutor History of Mathematics, William Oughtred
• MacTutor History of Mathematics, Edmund Gunter
• MacTutor History of Mathematics, John Horton Conway
• Internet Archive catalogue, The Trachtenberg Speed System of Basic Mathematics (Doubleday, 1960)
• D. Barner, G. Alvarez, J. Sullivan, N. Brooks, M. Srinivasan and M. C. Frank, Learning Mathematics in a Visuospatial Format: A Randomized, Controlled Trial of Mental Abacus Instruction, Child Development 87(4), 2016 (PubMed record)
• M. C. Frank and D. Barner, Representing exact number visually using mental abacus, Journal of Experimental Psychology: General 141(1), 2012
• D. F. Mansfield and N. J. Wildberger, Plimpton 322 is Babylonian exact sexagesimal trigonometry, Historia Mathematica 44(4), 2017
• E. Robson, Words and Pictures: New Light on Plimpton 322, American Mathematical Monthly 109(2), 2002

Further reading, cited in the text but not fetched for this post: Bharati Krishna Tirtha, Vedic Mathematics (Motilal Banarsidass, 1965); S. G. Dani, "'Vedic Mathematics': Myth and Reality", Economic and Political Weekly 28(31), 1993; Kim Plofker, Mathematics in India (Princeton University Press, 2009); Jakow Trachtenberg, The Trachtenberg Speed System of Basic Mathematics, translated by Ann Cutler and Rudolph McShane (Doubleday, 1960); Hang Young Pai, The Complete Book of Chisanbop (Van Nostrand Reinhold, 1981); E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways for Your Mathematical Plays (Academic Press, 1982), for the Doomsday rule.

Last reviewed: 25 October 2026.