2016 MCWC

The 2016 Mental Calculation World Cup was the seventh edition of the event. Jeonghee Lee (South Korea) finished first overall in the surprise tasks (and third overall). Yuki Kimura (Japan) won the overall championship, with Tetsuya Ono (Japan) finishing second overall.

Wenzel Gruss finished fifth overall (and sixth in surprise tasks). Freddis Reyes finished eighth, Sudhanshu Bhatia finished 15th, and Daniel Timms finished 16th.

Surprise Task Categories (2016)

The 2016 MCWC surprise task category included the following five task types:

• (A × B) − (C × D),
  where A and D were three-digit numbers and B and C were two-digit numbers

• Area calculation of an intermediate-level geometric figure,
  computed to as many significant digits as possible (with the value of π given)

• A 5 × 5 multiplication where the answer was given, and the competitor had to find the middle digit of the first number

• Solving x^3 − 5x^2 = 2016,
  to as many significant digits as possible

• Various calculations mixing multiplication, division, and exponents between 2 and 6,
  with the answer always being an integer

Challenge Task

The challenge task for 2016 was:

• 2016^(1/4) − 2016^(1/6) + 2016^(1/7)

The answer was to be calculated to as many significant digits as possible.

Notes on Calculation League Coverage

As of this writing, the Calculation League category for 2016 MCWC contains:

• the challenge task,

• the first surprise task, and

• two different versions of the final surprise task.

Other versions of the final surprise task are likely to be added later.

Example 1

(845^2 × 93^3) / 195^2

(Easy Level)

Although this question may have been generated from the “easy” category, it is not easy and would require significant experience to solve within the 30-second time limit.

I would begin by cubing 93.

Computing 93^3

We can calculate this directly:

• 93^2 = (93 + 7)(93 − 7) + 7^2

• 100 × 86 + 49 = 8649

Now multiply:

• 8649 × 93 = 804357

Alternatively, we could use the binomial theorem:

• 90^3 + 3·90^2·3 + 3·90·3^2 + 3^3

• 729000 + 72900 + 2430 + 27 = 804357

Reducing (845 / 195)^2

Now we must deal with:

• (845 / 195)^2

Computing 804357 × (845 / 195)^2 directly would be extremely difficult. Instead, we should focus on factoring and reduction. In this case, the number generation is favorable:

• 845 / 195 = 169 / 39 = 13 / 3

At this point, there are several possible approaches, depending on skill level.

Multiplying by (13 / 3) Twice

One option is to multiply 804357 by 13/3 twice, directly.

This would involve:

• quadrupling 804357 → 3217428,

• adding 804357 / 3 → 268119,

• giving 3485547.

We would then have to repeat the process again, quadrupling 3485547 and adding 3485547 / 3.

More Efficient Approach

A more effective approach is to multiply 804357 by (169 / 9).

We can immediately see that 804357 is divisible by 9, since its digits sum to 27.

• 804357 / 9 = 89372

Now we need to compute:

• 89372 × 169

By dividing first and multiplying second, we perform:

• a 6-digit ÷ 1-digit division, and

• a 5 × 3 multiplication.

If we did this in the opposite order, we would need:

• a 6 × 3 multiplication, and

• a 9-digit ÷ 1-digit division, which is noticeably more difficult.

In a competition setting, once we reach 89372 × 169, it is permissible to work right to left, performing 5 × 1 multiplications and typing each digit as you go.

Example 2

5898^(1/4) − 5898^(1/6) + 5898^(1/7)

(Advanced Level)

This question would be extremely difficult to calculate without logarithms.

If several significant digits are required, using logs may be appropriate, since we only need to estimate the logarithm of a single base (5898), followed by simple divisions.

Logarithmic Estimation

We estimate:

• log(5898) = 3 + log(5.898)

Linear interpolation is most inaccurate between 1 and 2, so we estimate:

• 3 + log(5) + (0.898 × (0.778 − 0.699))

• 3 + 0.699 + 0.071 ≈ 3.77

Now divide 3.77 by 4, 6, and 7:

• 3.769 / 4 → 0.94225

• 3.769 / 6 → 0.628125

• 3.769 / 7 → 0.53843

Interpolating Back

Using reference values:

• log(9) → 0.954

• log(8) → 0.903

• log(5) → 0.699

• log(4) → 0.602

• log(3) → 0.477

We arrive at the estimates:

• 5898^(1/4) → 8.77

• 5898^(1/6) → 4.27

• 5898^(1/7) → 3.49

This gives:

• 8.77 − 4.27 + 3.49 → 7.99

(The correct answer is 7.97.)

Alternative Estimation Path

Another approach is to start with square roots.

• √5898 → 76.7984

Then:

• √76.7984 → 8.76

• ∛76.798 → 4.25

If we estimate the 7th root as halfway between 4.25 and √8.76 (≈ 2.96), we obtain 3.6.

This leads to an estimated answer of 8.1.

However, the 7th root of 5898 is closer to the 8th root than the 6th root, so taking the midpoint slightly underestimates the value. By adjusting downward, we can obtain additional significant digits.

Example 3

((87 × 54)^2) / 4^6

(Expert Level)

It is helpful to restructure this expression:

• (87 × 54)^2 / (4^3)^2

• (87 × 54)^2 / 64^2

• (4698 / 64)^2

Computing the Square

We begin with the division:

• 4698 / 64 = 73.4 (or 73.406)

We can now square this value.

If we square 73.4 directly:

• (734 + 34)(734 − 34) + 34^2

• 538756

Moving the decimal two places to the left gives:

• 5388, which is correct.

Higher Accuracy Option

If more significant digits are needed, we can use the expansion of (a + b)^2:

• 70^2 + 2·70·3.406 + 3.406^2

• 4900 + 476.84 + 11.6 → 5388.44

This value is correct within 0.03.