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Find the Difference

Stage: 1 Challenge Level: Challenge Level:2 Challenge Level:2

Many of you sent us answers for this problem but very few gave us details about how you solved it. Please do try to explain how you reach your solutions.

Stephen, who is home-educated, told us:

I thought about all the possible subtractions and eliminated all the subtractions that have the same number repeated in them. (e.g. 6 - 3 = 3). That's a good way to start, Stephen. It's a shame that you didn't say what you did next.

However, the Junior Class at Ysgol Bryncrug in Gwynedd, Wales sent an extremely well-explained solution:

We thought which number to put on the top. We started with 1. That meant 2, 3 or 3, 4 or 4, 5 or 5, 6 could go under it.
We tried 2, 3 that left us with 4, 5, 6 for the bottom line. None of these gave us a difference of 3.
Then we tried 3, 4 and this left us with 2, 5, 6 which worked.
Then we tried 4, 5 and this left us with 2, 3, 6 for the bottom. None of these gave a difference of 5.
Then we tried 5, 6 on the second line and it is impossible to make a difference of 6 because it is the highest number.

So only 3, 4 works under 1:

first solution

We then saw that we could swap 3 and 4 around if we also swapped 5 and 6:

second solution
Then we tried 2 at the top. Working the same way we found out that only two ways could work:

third and fourth solutions

Then we tried 3 at the top. Working the same way we found out that four ways could work:

fifth, sixth, seventh and eighth solutions

Then we tried 4 at the top. On the second line we would have to use 1, 5 or 2, 6.
1, 5 didn't work out because we can't make a difference of 5 on the bottom line (6+1) because we have used 1 in the middle line.
2, 6 didn't work because we can't make a difference of 6 because it is the biggest number and it must go on the bottom line.

When we tried 5 on the top we have to use 1, 6 in the middle and we can't make a difference of 6 on the bottom.

There wouldn't be a point for us to try 6 at the top because no two numbers can give us a difference of 6 because it is the highest number.

So we think there are 8 solutions.

What fantastic logical reasoning, well done. I like the way you have gone through each possible top number in order. Having a system like that ensures that you find all the solutions.