2024 MCWC

Written By Sam Engel

The 2024 Mental Calculation World Cup was the 10th (and most recent) edition of the event. Aaryan Shukla (India) was the overall winner of the surprise tasks and also the overall champion, winning all four standard tasks.

Kaloyan Geshev (Bulgaria) finished second overall and second in surprise tasks, while Hiko Chiba (Japan) finished third overall.

Other GMCA members or Calculation League participants with notable results include:

  • Samuel Engel (USA) — 4th overall (5th in surprise tasks)

  • Mishti Shah (India) — 5th overall

  • Swanand Bhore (India) — 6th overall

  • Freddis Reyes (Cuba) — 7th overall

  • Domenico Mancuso (Italy) — 8th overall (6th in surprise tasks)

  • Veer Bagi (India) — 9th overall

  • Boris Quach (France) — 11th overall (8th in surprise tasks)

  • Hua Wei Chan (Malaysia) — 13th overall (10th in surprise tasks)

  • Daniel Timms (UK) — 17th overall

Surprise Task Categories (2024)

The 2024 MCWC surprise task category contained the following five tasks:

  1. A variety of formats involving the addition and subtraction of products of two two-digit numbers

  2. (a/b) + (c/d) + (e/f) = x/g, where the competitor had to find x

    • a through g were one-, two-, or three-digit numbers

  3. 1985^.5 / pi, to as many significant figures as possible (with pi given)

  4. (a × b) − (c × d), with each number being five digits

  5. Find the largest result among three values:

    • the 2nd / 3rd / 4th / 5th root of a four-digit number

    • a one-digit number (to three decimal places) raised to the 2nd / 3rd / 4th / 5th power

    • a / b, where b is a three-digit number and a is a four- or five-digit number

As of Season 3, Calculation League omits the third surprise task. The final surprise task is modified to fit the Calculation League format.

Example #1

4.025^4 − 67488 / 666

(Easy Level)

We only need to solve this question to the nearest integer.

The division can be computed easily:

  • 67488 / 666 = 101, with a remainder of 222

So the full value is 101 1/3, although additional digits are unnecessary here.

To compute 4.025^4, the Binomial Theorem is efficient.

First square:

  • 4.025^2 = (4 + 0.025)^2

  • = 4^2 + 2·4·0.025 + 0.025^2

  • = 16 + 0.2 + 0.000625

Unless a high number of digits is required, ignoring the final term is appropriate.

Now square again:

  • 16.2^2 = 16^2 + 2·16·0.2 + 0.2^2

  • = 256 + 6.4 + 0.04

  • = 262.44

Subtracting:

  • 262 − 101 = 161

Alternatively, instead of applying (x + y)^2 twice, we could expand (x + y)^4 directly, with x = 4 and y = 0.025:

  • 4^4 + 4·4^3·0.025

Even just the first two terms already give 262.4, which is sufficiently accurate for this question.

Example #2

(76×13) + ((84×18) + (52×25) − (95×94)) − (48×99)

(Advanced Level)

For this question, the order of calculation is not especially important. My preference is to minimize the size of the numbers I am holding at each step.

One efficient approach:

  • 48×99 = 4752

  • 52×25 = 1300 → 4752 − 1300 = 3452

  • 76×13 = 988 → 3452 − 988 = 2464

  • 84×18 = 1512 → 2464 − 1512 = 952

  • 95×94 = 8930 → 8930 + 952 = 9882

Final answer: −9882

General Strategy Notes

For this type of question, I generally recommend:

  • Reserve the most difficult calculation for last, so you can type or write while finishing it

  • Start with the second-most difficult calculation

  • If both positives and negatives are present, balance them early to reduce the number of digits you must keep

  • At each step, do the next most difficult calculation

Digit-by-Digit Alternative

Alternatively, this can be done digit-by-digit, which may be useful for larger expressions. It does require practice, especially when switching between different cross-multiplications.

Working right-to-left:

  • (4×5) + (8×9) − (2×5) − (4×8) − (6×3) = 32
    → Type 2, carry 3

  • 3 + (5×9) + (4×9) + (8×9) + (4×9) − (2×2) − (5×5) − (4×1) − (8×8) − (3×7) − (6×1) = 68
    → Type 8, carry 6

  • 6 + (9×9) + (4×9) − (5×2) − (8×1) − (7×1) = 98
    → Type 98

Example #3

984983^(1/7) − 6392136 / 94329

(Expert Level)

For questions of this type, it is helpful to estimate the order of magnitude of each component before computing.

We know:

  • 6392136 / 94329 is between 10 and 100

  • 984983 < 10^7, so 984983^(1/7) is less than 10

Calculating the division:

  • 6392136 / 94329 ≈ 67.8

Because we can compute the second component to three significant digits, we only need to estimate the first component to within about 0.5 to ensure correctness to the nearest integer.

We know:

  • log(1,000,000) = 6

  • log(984,983) is slightly less than 6

Dividing by 7:

  • 6 / 7 ≈ 0.86

So the first term is approximately:

  • 10^.86

Since:

  • log(7) ≈ 0.845

We conclude that 984983^(1/7) is slightly greater than 7.

Therefore, the final estimate is:

  • 7 − 68 = −61

Sam Engel
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2022 MCWC