2012 MCWC

The 2012 Mental Calculation World Cup was the fifth edition of the event. Japanese competitor Naofumi Ogasawara won all five surprise tasks, as well as addition and square roots, and also won the overall championship.

GMCA panelist Hua Wei Chan finished second overall and fourth in the surprise tasks. Jan van Koningsveld (Germany) finished third overall. Freddis Reyes finished 11th overall, and Daniel Timms finished 18th overall.

Surprise Task Categories (2012)

The 2012 MCWC surprise task category included the following five task types:

• Mixed addition / subtraction (six-digit numbers).
  The answer could be positive or negative, and the format was fixed as:
  a + b − c − d + e + f − g − h

• a² + b² = c², with b to be found

• Converting meters to feet and inches
  (meters given as a two- or three-digit number)

• (a / b) + (c / d) + (e / f) compared to g
  (greater than or less than g).
  Values a through f were two- or three-digit numbers (with one four-digit number), and g was a two- or three-digit number.

• (a × b) / (c × d)
  with a and b being three- or four-digit numbers, and c and d being two- to four-digit numbers.
  The answer was an integer.

Notes on Calculation League Modifications

The 2012 MCWC also included a cube-root competition, and cube roots are included in the 2012 MCWC tasks for Calculation League.

Otherwise:

• Calculation League does not use integer answers for the final category, and

• In the fourth category, the question is restructured as a fraction-addition problem.

Example 1

914427 + 105787 − 856277 − 350971 + 709062 + 362172 − 762164 − 870404

(Easy Level)

In a mixed addition/subtraction question where the answer is not guaranteed to be positive — and where there are enough columns that the problem is likely to be broken into parts — you should start from left to right and determine whether the final answer is positive or negative.

Mental calculation rules generally prohibit typing or writing intermediate results. It is debatable whether assuming the answer is positive and then flipping the sign later violates these rules. This approach is closer to submitting an incorrect answer than writing an intermediate one; however, it is still not in the spirit of the rules.

For isolated occurrences, it is unlikely to be addressed, but if a competitor uses this as a regular strategy, it will result in point deductions.

Regardless, it is not an efficient approach.

Determining the Sign

In this case, we start from left to right and work with the number of columns we are most comfortable with. I would choose two columns.

This gives:

• 91 + 10 − 85 − 35 + 70 + 36 − 76 − 87 → −76

Now we know the final answer is negative, and we can finish the calculation from right to left. We should remember −76, so this step does not need to be repeated.

Completing the Calculation

Now take the final two columns:

• 27 + 87 − 77 − 71 + 62 + 72 − 64 − 4 → 32

Here, we type 68 (100 minus 32) and carry 1.

Next:

• 1 + 44 + 57 − 62 − 9 + 90 + 21 − 21 − 4 → 117

We type 83 (200 minus 117) and carry 2.

Finally:

• 2 + (−76) = −74

Final answer: −748368

Example 2

(773² − 372²)^.5

(Advanced Level)

Algebra provides a useful shortcut for computing the difference of two squares:

a2−b2=(a+b)(a−b)a^2 − b^2 = (a + b)(a − b)a2−b2=(a+b)(a−b)

Applying this identity:

• (773 + 372)(773 − 372) → 1145 × 401

Compute:

• 1145 × 400 = 458000

• Add 1145 → 459145

Now apply basic square-root estimation:

• 459145^.5≈677.6

Note that the difference between 6772677^26772 and 6782678^26782 is:

• 677 + 678 = 1355

This means that, to obtain the answer to the nearest 0.5, we do not need the last three digits of 459145. Using 459000 is sufficient to guarantee correctness to the nearest 0.5.

Example 3

(58237 × 75220) / (50798 × 41126)

(Expert Level)

For this final category, the ratio between the numbers in the numerator and denominator determines the difficulty of the problem. In this case, we effectively have a relatively easy version of the fraction-multiplication category.

The correct answer to this question is:

• 2.09685905292

Estimation Approaches

Using one-digit approximations:

• (6 × 7.5) / (5 × 4) = 45 / 20 → 2.25

Using two-digit approximations:

• (58 × 75) / (51 × 41) = 4350 / 2091 → 2.08

At this point, we are already close enough to the required answer of 2.1.