## Professor Shanku posed the puzzle of cats and mice to his friends and none could find the answer

While taking evening tea with his friends and discussing growing menace of mice in the neighborhood, Professor Shanku spoke, "If a number of cats killed 999,919 mice between them and each cat killing equal number of mice, how many cats were there?" One friend suggested that perhaps 999,919 cats killed one mouse each, but Shanku promptly reminded that he told "mice" not "mouse". Another suggested that possibly 1 cat killed all the mice, but again Shanku reminded that he had told "cats", not cat. For helping his friends, he gave a hint that each cat killed more mice than the number of cats. "Now find the correct answer." He ended.

**Time for you to solve:** 10 minutes, a generous amount of time.

The discussion among the friends duly created just an amount of confusion. To get to the job of unraveling the mystery, the *barebones facts and conclusions must be made without any confusion.*

### Solution to the puzzle on cats and mice

Facts:

- 999,919 mice were killed.
- Each cat killed equal number of mice.
- The number of mice killed by each cat was larger than the number of cats.

That means if 999,919 were divided by the number of cats we'll get a whole number, the number of mice killed by each cat.

In other words, we just have to factorize the number 999,919 into a two factor product numbers. The smaller product will be the number of cats. A sigh of relief.

But hold on. *How to factorize the gigantic number of 999,919!*

At last the detestable, to be avoidable trial and error method to be taken up? But that will take a very long time.

Most importantly, no puzzlist worth its name would leave the puzzle to solve with no option other than random trial error method. **A basic characteristic of a puzzle is:**

A puzzle must have a hidden clue.This clue leads to the solution.

Taking heart, the number 999,919 is given a real close look. What is this peculiarity of the number? **The number is less than 1 million by just 81.**

This sparks real interest. Aren't 81 and 1 million both squares?

Yes, of course. And, 999,919 = 1000^{2} - 9^{2} = (1000+9)(1000-9) = 1009 x 991.

The most basic algebraic relation, (a^{2} - b^{2}) = (a + b)(a - b) just cannot be forgotten.

Solution happened quick with the revelation of the hidden code:

The number of cats were 991, each of which killed 1009 mice for a total of 999,919 mice.

**Still a lingering doubt:** Are these two numbers 1009 and 991 prime numbers? If not, there will be more than one answer, but the Professor mentioned just one answer.

Relief. A quick check established both 991 and 1009 as prime numbers. Answer is only one.

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**Matchstick puzzles**You may also look at the full collection of puzzles at one place in the **Challenging brain teasers with solutions: Long list.**

*Enjoy puzzle solving while learning problem solving techniques.*