2006 MCWC

The 2006 Mental Calculation World Cup was the second edition of the event. As in 2004, there were only two surprise task categories, and there were no reported results from those categories. The overall event was again won by Robert Fountain (UK).

Surprise Task Categories (2006)

The 2006 MCWC surprise task category included the following two task types:

• A / B ± C / D, with one- to three-digit numbers

• Find the next prime, where all numbers were between 1000 and 1700

Calculation League does not currently use the “next prime” task.

Notes on Task Format

While the posted MCWC task sheets do not include explicit instructions, it is reasonable to assume that MCWC required answers to be written in fraction form, possibly in reduced form.

Calculation League does not currently require — or support — writing answers in fraction form. As a result, these questions are treated simply as problems involving two basic divisions, followed by either addition or subtraction.

General Strategy Remarks

There is unlikely to be any meaningful special strategy for these questions. They are essentially an easier version of standard fraction addition/subtraction tasks.

In practice, a competitor can choose between two reasonable approaches:

• (a) Perform each division to the required number of digits and then adjust the answer as needed; or

• (b) Perform each division to one additional digit, which significantly increases the probability of submitting a correct answer on the first attempt.

Example 1

(238 / 793) − (248 / 586)

(Easy Level)

On this first question, I immediately notice that the first term is almost exactly 0.3, since:

• 3×793=23793 × 793 = 23793×793=2379

For the second term, completing the division leads to a value near 0.42. After calculating the second digit, I can determine that:

• (a) the value lies between 0.42 and 0.43, and

• (b) it is closer to 0.42 than to 0.43

Because it is closer to 0.42, we obtain:

• 0.30−0.42=−0.120.30 − 0.42 = −0.120.30−0.42=−0.12

So the answer is −0.12.

Example 2

(7843 / 662) + (9014 / 9103)

(Advanced Level)

In this question, because the first term exceeds 10, we only need the answer to the nearest integer.

It is clear that the second term is 1.0 when rounded to the nearest 0.1.

Unless the first term is very close to a halfway point, performing the first division to the nearest integer and then adding 1 will produce a correct result.

• 7843÷6627843 ÷ 6627843÷662 is approximately 12

Therefore, an acceptable answer to the question is 13.

Example 3

(74504 / 65158) + (17551 / 34372)

(Expert Level)

In this example, rather than performing a precise division for the second term — which has a five-digit denominator — I first notice that the quotient is just slightly above 0.5.

Doubling 17551 gives:

• 2×17551=351022 × 17551 = 351022×17551=35102

This is 730 more than 34372. Half of 730 is 365, which tells us that the division will result in:

• 12+36534372\frac{1}{2} + \frac{365}{34372}21​+34372365​

In other words, the value is just greater than 0.51.

Now consider the first term:

• Performing the division gives approximately 1.14

At this point, I can enter 1.65 as the answer, or I can compute one additional digit of the first quotient to confirm that the final result is closer to 1.7 than to 1.6.