I’m thinking of a 4-digit number. If I add its four digits together I get 34. How many different numbers are there that I could possibly be thinking of?
Solution
(i) It’s first important to establish what the four digits could be. What four digits when added together give a total of 34?
Well, suppose that one of the digits is 6. Then the other three add to 34 – 6 = 28. But 3 × 9 = 27, so no three digits exist that can add to 28. So none of the original digits is 6. In fact none of them is smaller than 6 for the same reason.
Now suppose that one of the digits is 7. Because the other digits add up to 27, they must all be 9s. So here I had to have 7, 9, 9, 9.
Now suppose that one of the digits is 8. Because the other digits add up to 26, it can easily be seen that one is 8 and the others are 9s. So here I had to have 8, 8, 9, 9.
Clearly they are not all 9 (4 × 9 = 36 > 35).
(ii) So what numbers can be made up with one 7 and three 9s?
There are just four of them. They are 9997, 9979, 9799, 7999.
(iii) So what numbers can be made up with two 8s and two 9s?
There are just six of them. They are 9988, 9898, 9889, 8998, 8989, 8899.
(iv) The answer to the question is therefore 10.
Extension
Make up some problems like this to try on your friends. You could start off with 2- or 3-digit numbers.