Challenge Level

After playing this game several times, we suggested that you might like to search for some impossible pairs of cards as well as numbers which satisfy three or even four cards.

We received a solution from Ariel, who sent in a list of several pairs of impossible cards.

Ariel suggested that the the 'Square number' and 'Prime number' cards are an impossible pair. Is she correct? You might find it helpful to think about some actual numbers to help to convince yourself about your answer. What do you notice about the factors of square numbers such as 25 and 49? What's different between them and the factors of prime numbers like 13 and 23?

She also suggested that 'Two less than a multiple of 10' and 'One more than a multiple of 5' were an impossible pair. Again, it might be helpful to think about some actual numbers to help you to decide whether she is correct. What do you notice about a list of numbers which are two less than a multiple 10? Now consider listing some numbers which are one more than a multiple of 5. When you compare your two lists, what do you notice?

Here's the rest of Ariel's list of impossible pairs:

'The product of two primes' and 'A prime number.'

'The product of two primes' and 'A square number'

'Multiple of 3 but not multiple of 9' and 'Digits sum to 9'

'Odd number' and 'Even number that is not a multiple of 4'

'Odd number' and 'Two less than a multiple of 10'

'A multiple of 7' and 'A factor of 24'

'A multiple of 7' and 'A Factor of 60'

Are they all impossible pairs? Can you explain your reasoning?

We also asked about any numbers which satisifed three or four cards. Ariel suggested the following examples:

9 satisfies 3 cards, which are a square number, digits sum to 9 and an odd number.

0 also satisfies 3 cards, which are one less than a square number, a square number and a multiple of 7. Since 0 also satisfies the requirements of a triangular number (but this is argued), you can say that it satisfies 4 cards too.

2 satisfies 4 cards, which are a prime number, an even number but not a multiple of 4, a factor of 24 and a factor of 60.

Ariel also explored whether there is a number that satisfies none of the cards:

Yes, there is. 32 is the smallest number. To find a number like this, we can first consider about the ones digit. The digit can neither be odd, nor it can be 1, 6 or 8. So the number must have its ones digit being 2, 4 or 0. Next, the number cannot be a multiple of 3 or 7, but it must be a multiple of 4. Then, the number is better to be larger than 60, to avoid it being a factor of 24 or 60. What is remaining to test is whether it is a square number, a triangular number, or one less then a square number. These numbers are not considered. Then the remaining numbers are what we want to find. Using this method, we can find that the numbers that have this feature in 100 are 32, 40, 44, 52 and 92.

Well done finding all these numbers. Ariel! Are there any other possible answers? How do you know?