100th Mathematical
Puzzle Financial Daily

The amount of data around us doubles every year. With this enormous volume of data, not only the amount of valuable information is growing, but also the amount of noise surrounding it. Distilling the truly important information and discovering the real patterns is the challenge arising due to this.

To do so we can use various advanced modelling techniques and machine learning algorithms. However, these methods are no magic solution. To solve difficult business challenges, the use of old puzzle skills should not be underestimated. These can be used to split seemingly impossible problems into smaller questions which are easier to solve, or to find a generic pattern across different situations.

That’s exactly why we as MIcompany love mathematical puzzles, riddles, and brain teasers. To share our hobby with a larger audience, and to show the importance of these puzzle skills, a cooperation with the Dutch newspaper ‘Financieel Dagblad’ (Financial Daily) was started. As of September 2015, a new mathematical puzzle has been posted weekly in the weekend edition ‘Morgen’. After 99 publications, this weekend the 100th brain teaser was published. We hereby invite you to join us and to challenge yourself by solving this anniversary puzzle.

#### Anniversary Puzzle

Boukje is inspired by all previous mathematical puzzles in the Financial Daily and she feels that the standard operators, such as +, -, * and / are quite boring. That’s why she invents a new operator ¤, for which the following equations hold:

1 ¤ 4 = 5
2 ¤ 5 = 12
3 ¤ 6 = 21
4 ¤ 7 = 32
5 ¤ 8 = 45

Can you discover an operator that gives these results? And can you find integers a and b for which a ¤ b = 100? If yes, for how many pairs (a, b) is it true that a ¤ b = 100, given that a and b should both be non-negative integers?

There are multiple operators that give these results. The simplest one is the following:

a ¤ b = a + a * b = a * (b+1)

For example: 1 ¤ 4 = 1 + 1 * 4 = 5 or 5 ¤ 8 = 5 + 5 * 8 = 45.

Whether there are pairs for which a ¤ b = 100, depends on the chose operator. Here we give the answer for the operator above.

For this operator there are 9 pairs that give 100 as a result: (1, 99), (2, 49), (4, 24), (5, 19), (10, 9), (20, 4), (25, 3), (50, 1), (100, 0).

To find these pairs, we write our formula even simpler:

a ¤ b = a * (b+1) = a * c

We can find all pairs (a, c) for which the result is 100, by looking at the prime factorization of 100:

1. Prime factorization of 100 = 2 * 2 * 5 * 5
2. We can write 100 as a product of a and c by dividing these 4 prime factors over two terms. For example: a = 2 * 2 and c = 5 * 5 à a * c = 4 * 25 = 100
3. For a we can choose 0, 1 or 2 times 2 and 0, 1 or 2 times 5. The other factors we put in c. This gives us 3 * 3 = 9 correct pairs: Hungry for more challenges? Go to the website of the Financieel Dagblad or order our puzzle book Puzzalytics.