My son who is at primary school was given the following homework:

"Finding all possibilities Logic Problem"

Jessica and Simon were blowing up balloons for Garreths birthday. There were at least two of each. Some balloons had three spots and some had 5 spots.

There were 31 spots altogether.

Q: How many balloons had three spots and how many had 5 spots?

What if there were 24 spots?

What if there were 65 spots?

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OK, so knowing that there were two of each to start with (5+5+3+3=16 spots) we only need to establish the possible permutations for the remaining 15 ie 3 x 5 spot balloons or 5 x 3 spot balloons

For 24 spots there is only one possible answer (24 - 16 = 8 spots = 1 x 5 spot balloon and 1 x 3 spot balloon

For 65 there are a few permutations 65-16=49

8 x 5 + 3 x 3 = 49

2 x 5 + 13 x 3 = 49

5 x 5 + 8 x 3 = 49

But can we be sure that we have found all the possible answers? Is there a formulae for testing and am I posting this question in the right place!