2018 MCWC

The 2018 Mental Calculation World Cup was the eighth edition of the event. In the surprise task category, the top performance was by Wenzel Gruss (Germany), while Samuel Engel (USA) finished third.

Surprise and Challenge Tasks (2018)

At the 2018 MCWC, there were five surprise tasks and one challenge task.

One surprise task involved determining whether a given A × B = C was correct, with A and B being nine-digit numbers. This task is substantially different in nature from Calculation League questions and is not represented.

A second question format consisted of standard mod questions, which are already represented in the mod category.

A third question involved converting units of measurement (for example, time and distance). The calculations required for these questions were relatively simple, and as a result, this question format has also been omitted from Calculation League.

Tasks Included in Calculation League

As a result of the above omissions, only three of the six non-standard tasks from the 2018 MCWC are included in Calculation League. These are:

• (a × b) + (c / d)
  (challenge task)

• a / (b × c × d)
  (won by Wenzel Gruss at MCWC 2018)

• (a / b) − (c / d) + (e / f) + (g / h) + (i / j)
  (won by Samuel Engel at MCWC 2018)

As usual, Calculation League removes any “exact” answer limitations. Instead, the numbers are random, and the divisions are unlikely to result in integers. For the third question format, the placement of plus and minus signs between fractions may also change.

Example 1

(510 × 406) / (83275407 / 465)

(Easy Level)

The easy-level version of this task is almost identical to the challenge task format used at MCWC. The only differences are that:

• the value of c / d is unlikely to be an integer, and

• the result may be a five- or six-digit number.

Choosing an Order of Operations

In general, I would approach this question by performing the division first, since division is typically done left to right. Because intermediate results are easier to remember when working left to right, this approach allows more flexibility in how the multiplication is performed afterward.

In this case, however, the numbers in the multiplication are particularly “nice” — they include two zeros and a one. Because of this, I would choose to perform the multiplication first, using either standard or cross multiplication.

Even without calculating explicitly, we can see that because of the digits in a (a five, a one, and a zero) and the fact that all digits in b are even, the multiplication will be straightforward and the result easy to remember.

Computing the Multiplication

We compute:

• 510 × 406

This can be written as:

• (500 × 406) + (10 × 406)

• 203000 + 40600 → 243600

Remembering the chunks “24”, “36”, and “00” should not be particularly difficult.

Computing the Division

Moving to the division, note that because the first digit of the numerator (8) is larger than the first digit of the denominator (4), the answer will be six digits.

At this point, we can use either standard division or cross-division to obtain the result:

• 179087

For competition purposes, I would recommend typing the answer as you calculate, though this takes practice. Especially when the first result (243600) is easy to remember, it may be less risky to compute the nearest integer for the division and then combine the results afterward.

Practicing writing or typing while continuing to calculate can interrupt the calculation process when you are not used to it, but it is an important skill emphasized by some top competitors (particularly in Japan).

Example 2

3411247 / (97 × 52 × 86)

(Advanced Level)

In this question, we only need two significant digits of accuracy. I would recommend approaching the calculation in the following order.

Step 1: Multiply 52 × 86

• 50 × 86 = 4300

• 2 × 86 = 172

So:

• 52 × 86 = 4472

Step 2: Multiply by 97 (Estimate)

Since 97 is close to 100, we can estimate:

• 4472 × 100 → 447200

• 4472 × 3 → 13416

We do not need full accuracy here.

Step 3: Estimate the Final Quotient

Using the first three digits:

• Numerator: 341

• Denominator: 43.4

We test:

• 8 × 43.4 = 347.2 (too large)

Now subtract multiples of 4.3:

• 7.9 → 342.9

• 7.8 → 338.6

Since 341.1 is closest to 7.9, the answer is 7.9 to two significant digits.

Example 3

(8013 / 7123) − (8111 / 3456) + (2210 / 3342) + (9056 / 7590) + (9860 / 5595)

(Expert Level)

For this task, we effectively have a relatively easy version of the fractions addition category. In the original 2018 MCWC version, it is almost always the case that only two significant digits are required.

In this format, we simply compute each division.

Estimating Each Term

• 8013 / 7123 → about 1.1 (rounded down)

• 8111 / 3456 → about 2.35

• 2210 / 3342 → 0.66

• 9056 / 7590 → 1.2 (rounded up)

• 9860 / 5595 → 1.7 (rounded down)

Adding these estimates gives:

• 2.31, which is correct to two significant digits.

Notes on Rounding Strategy

When estimating, I prefer to estimate to the same level of accuracy required in the final answer.

Some additional remarks:

• If a value is very close to halfway between two numbers (for example, 2.35), estimating halfway is appropriate. This can later be balanced by adjustments elsewhere.

• For the third calculation (2210 / 3342), I estimated an additional digit (0.66 instead of 0.70) because it was easy to do and provided extra information about whether to round up or down.

• I recommend keeping track of whether you rounded up or down, so long as it does not substantially interfere with the calculation. If the difference between the number of upward and downward roundings is two or more, an adjustment may be necessary.

Here, I rounded up once and down twice, so no adjustment is needed. However, this suggests that the true value is likely slightly larger than 2.31 (it is 2.34).

Effect of Trailing Digits

One important observation for these questions is the limited effect of trailing digits.

• The answer to five significant digits is 2.3465.

If we use only one digit for each number:

• (8/7) − (8/3) + (2/3) + (9/7) + (9/5) = 2.2286

Using two digits:

• (80/71) − (81/34) + (22/33) + (90/75) + (95/55) = 2.3383

This shows that it is not even necessary to consider the third and fourth digits of the numbers. With informed estimation techniques, the problem can be solved using only one digit per number.