2022 MCWC
The 2022 Mental Calculation World Cup was the 9th edition of the event. In the surprise task category, the top two performances were by Japanese competitors Ono Tetsuya and Naota Higa. Other noteworthy performances included:
3rd — Kaloyan Geshev
5th — Samuel Engel
8th — Domenico Mancuso
9th — Aaryan Shukla
Surprise Task Categories (2022)
The 2022 MCWC category contained five different types of tasks:
Two tasks involving x / (y^.5)
In one version, x = 1000 (this task was won by Aaryan Shukla)
In the other version, x is variable
3A + 4B + 5C + 6D + 7E
(this task was won by Samuel Engel)(a^b) / (c^d)
(a × b × c)^.5, resembling the Multiplication / Roots category
(this task was won by Kaloyan Geshev and Domenico Mancuso)
Notes on Calculation League Modifications
After a few modifications, the Calculation League versions of these questions are generally more difficult than at MCWC:
In the second category, the expression is formatted horizontally instead of vertically, making rapid digit-by-digit calculation more difficult.
In the third category, competitors must calculate the result, rather than merely compare two values.
In the fourth category, MCWC questions had integer answers and were designed to permit factoring; this restriction is removed in Calculation League.
Example #1
929 / (286^.5)
(Easy Level)
We only need to calculate the above result to the nearest integer (or two significant digits).
In general, if seeking a correct answer on the first attempt, calculating three significant digits of the square root would be appropriate. Here, note that:
17^2 = 289
Therefore, 286^.5 is slightly less than 17.
This means the answer is slightly greater than 929 / 17. Computing:
929 / 17 = 54.647
This leads to a rounded answer of 55, which is correct to the nearest integer.
Accuracy via Estimation
We can confirm correctness using simple estimation techniques.
First, note that the difference between 16^2 and 17^2 is the sum of the two numbers:
16^2 → 256
17^2 → 289
Difference → 33
Since 286 is 3 less than 289, linear interpolation gives:
17 − (3 / 33) ≈ 16.91
This already gives four significant digits of accuracy; three would be sufficient.
Second, observe that:
929 / 16.9 = (929 / 17.0) × (17.0 / 16.9)
The factor 17 / 16.9 is equivalent to adding roughly 1 / 169 to the result.
Since 929 / 17 = 54.647, adding between 0.3 and 0.33 still clearly yields 55 to two significant digits. With more precision, a third significant digit could also be obtained.
Example #2
1000 / 974611^.5
(Advanced Level)
This format is based on the challenge task at MCWC 2022. The original challenge task required eight significant digits of accuracy, making it perhaps the most difficult MCWC task ever presented. Only two competitors earned any correct points.
The current Calculation League version is far simpler — in fact, it is the easiest task in the 2022 MCWC category.
This question can be solved using quantitative reasoning, without any practiced square-root technique.
We note that:
900^2 < 974611 < 1000^2
So 974611^.5 lies between 900 and 1000.
This leaves only two plausible answers:
1.0 or 1.1
Because 974611 is much closer to 1000^2 than to 900^2, we can confidently conclude that the answer is:
1.0
Example #3
(1482 × 1945 × 3929)^.5
(Expert Level)
The MCWC version of this question was designed so that:
Competitors could collect repeating factors, and
The final answer would be an integer
For example, if a factor of 2 appears in two of the numbers, each number can be divided by 2, while keeping an external factor of 2.
Calculation League avoids questions that strictly test the use of a single restricted technique with limited practical utility. That does not mean the principle itself should be avoided — reducing four-digit numbers to three-digit numbers can be very helpful — but only when factors can be identified almost immediately.
Here, there are no repeating factors:
3929 is prime
1945 = 5 × 389
1482 = 2 × 3 × 13 × 19
So direct calculation is necessary.
This is an expert-level question. Multiplying three four-digit numbers, holding the result, and taking a square root is extremely difficult unless the numbers are unusually favorable. Taking square roots first and then multiplying is even more difficult.
The required answer is:
106420
Progressive Approximation
If each root is calculated only to the nearest integer:
1482^.5 ≈ 38
1945^.5 ≈ 44
3929^.5 ≈ 63
Then:
38 × 44 × 63 = 105336
(about 1% off — not close enough)
Adding one more significant digit:
38.5 × 44.1 × 62.7 = 106455
(about 0.03% off — still not sufficient)
Adding a second additional digit:
38.50 × 44.10 × 62.68 = 106421
(only 1 off)
However, this required multiplying three unseen four-digit numbers, which is extremely demanding.
Estimation via Adjustment
A rough estimate:
(1500 × 3900 × 1950)^.5 → 106806
(accurate within 0.4%, but still insufficient)
To refine this, we analyze rounding effects:
Rounding 1482 → 1500 and 3929 → 3900
Added amount:
18 × 1945 × 3929
Subtracted amount:
29 × 1482 × 1945
Approximating:
18 × 3929 ≈ 71000
29 × 1482 ≈ 43000
Net added:
(71000 − 43000) × 1945 ≈ 54,000,000
To compensate, adjust 1945 downward:
54,000,000 / (1482 × 3929) ≈ 9
So use:
1500 × 3900 × 1936
Now:
1936 × 1.5 = 2904
→ 2,904,000
Taking the square root:
(2,904,000 × 1936)^.5 ≈ 106422
(only 2 off)
This reduces the problem from three four-digit multiplications to two four-digit multiplications.
Further Reduction
Since the ones digits add to 10, we can apply:
(2904 × 1936) = (2420 + 484)(2420 − 484)
= 2420^2 − 484^2
This works because:
the ones digits add to 10, and
the difference between the numbers is less than 2 × 1000
In practice, a standard left-to-right multiplication would likely still be preferable, but this structure is worth noting.
The critical decision is how many digits to work with in an expression of the form (A × B × C)^.5. The correct approach depends heavily on context — required accuracy and time constraints.
Example #4
(3 × 96668808) + (4 × 65563512) + (5 × 21717994) + (6 × 65810781) + (7 × 22927653)
(Expert Level)
At MCWC, this question was formatted vertically, allowing for rapid scanning and digit-by-digit multiplication (and I won this task).
In Calculation League, the question is formatted horizontally.
Unless you are comfortable storing a 9- or 10-digit number in your head, scanning back and forth is unavoidable. Either left-to-right or right-to-left multiplication works. I recommend two-digit blocks if you are comfortable with 2×1 multiplications.
Working in blocks:
(3×8) + (4×12) + (5×94) + (6×81) + (7×53) = 1399, write 99, carry 13
(3×88) + (4×34) + (5×79) + (6×7) + (7×76) = 1369, write 69, carry 13
(3×66) + (4×56) + (5×71) + (6×81) + (7×92) = 1907, write 07, carry 19
(3×96) + (4×65) + (5×21) + (6×65) + (7×22) = 1197, write 1197
Final answer: 1197076999
Digit-by-digit computation is also possible, but it requires more scanning and more careful digit tracking.
Example #5
(24^4) / (5^6)
(Advanced Level)
This is another format where the Calculation League version is more difficult than the MCWC version, which only required a greater-than / less-than decision.
The difficulty varies with the number of required significant digits.
Option 1
Break the expression apart:
((24^4) / (5^4)) × (1 / 5^2)
→ ((24 / 5)^4) / 25
Compute:
4.8^2 = 23.04
23.04^2 ≈ 529
Now:
525 / 25 = 21
Option 2
Restructure midway:
((24^2) / (5^3))^2
→ (576 / 125)^2
Since:
575 / 125 = 4.6
Then:
4.6^2 ≈ 21
Base Adjustment Insight
We can also rewrite:
(25^4 / 5^6) × (24 / 25)^4
= 25 × (24 / 25)^4
Multiplying 25 by (24/25) four times results in one full reduction of 1, followed by three slightly smaller reductions — making it almost immediately clear that the answer is 21.