Earlier today I gave you these three problems from a Nick Berry tale DataGenetics blog. Nick died last week at the age of 55, which I wrote about in original post.

1. Non-zero characters

Write 1,000,000 as the product of two numbers; none of them contain zeros.

(You might be interested to know that 10 x 10 x 10 x 10 x 10 x 10 = 1,000,000)

The solution 15625 x 64

Since 10 = 2 x 5, then a million is 2 x 2 x 2 x 2 x 2 x 2 x 5 x 5 x 5 x 5 x 5 x 5. If we group the fives and twos together, we get the answer . If none of the numbers must have zero (meaning they cannot have 10 as a divisor), then none of the numbers can have both 2 and 5 as a factor.

2. Lucy’s secret number

You are at a party and overhear a conversation between Lucy and her friend. In conversation, Lucy mentions that she has a secret number under 100.

She also admits to the following information: “The number is uniquely described by the answers to the following four questions:”

Q1) Is a number divisible by two?
Q2) Is a number divisible by three?
Q3) Is a number divisible by five?
Q4) Is a number divisible by seven?

She then proceeds to whisper the answers to these questions to her friend. Unfortunately, due to the ambient noise at the party, all you’ll hear is the answer one from the questions. Knowing this one answer allows you to determine the secret number. The answer you hear is yes. What is Lucy’s secret number?

The solution 70

Since there are four questions and each answer can be yes or no, there are sixteen possible combinations of answers. Lucy said that the answers to the questions definitely determined her score. So we need to look for combinations that uniquely define a number.

Let’s start with the combination of No, No, No, No. This combination allows the numbers 11, 13 (and a few others), so we can eliminate it.

Next, let’s try No, No, No, Yes. This combination allows for 7, 49 (and a couple more), so we can eliminate that as well.

Looking through all the combinations, there are only two that fix a single number:

No No Yes Yes defines 35

Yes No Yes Yes defines 70

So the secret is 35 or 70.

However, you will find that both of these solutions have the same answers to Q2″divisible by three”Q3″divisible by five” and Q4 “divisible by 7”. Therefore, if knowing the answer to only one question determines the quantity, the question should be Q1″divisible by 2”. We are told that there is an answer “yes”therefore, Lucy’s secret number is 70.

3. Naughty math elves

I write whole numbers from 1-9999 (inclusive) on a huge blackboard. Each number is written once.

At night, the blackboard is visited by a series of naughty math elves. Each elf approaches the board, picks two numbers at random, erases them, and replaces them with a new number that is the absolute difference of the two erased numbers.

This vandalism continues throughout the night until only one number remains.

The next morning I go back to the board and find the only number on the board. Is the remaining number odd or even?

The solution An even number

This problem can be solved simply by applying parity. Quantity is either odd or even. At the beginning of the night, 9999 numbers are written on the board. Of these numbers, 5000 are odd and 4999 are even.

When the elf chooses a pair of numbers, there are three possible permutations. He can choose two odd numbers, two even numbers, or one of each.

• If he picks two odd numbers, the absolute difference between the two odd numbers will always be even. What happened was that the number of odd numbers left was reduced by two.

• If he picks two even numbers, the absolute difference between the pair of even numbers will also be even. The number of odd numbers remains the same.

• If he chooses an odd number and an even number, the absolute difference between the pair will be odd. We lost one odd number, but gained a new one, so the number of odd numbers remains the same.

From this you can see that either the number of odd numbers or remains the sameor halveswith every act of corruption.

The number of odd numbers on the board started at 5000, which is even and must remain even.

If we get down to one number, it is has be an even number.

I hope you enjoyed the puzzles. I’ll be back in two weeks.

Thanks to Ian Mercer for his help with today’s column.

Nick Berry Datagenetics The blog is a treasure trove of material and is highly recommended.

I post a puzzle here every two weeks on Monday. I’m always on the lookout for great puzzles. If you want to suggest one, write to me.