Can you figure out this pentagon puzzle?

Can you figure out this maths puzzle? Photo / Getty Images

Mathematical genius Adam Spencer is the author of a new book The Number Games. Spencer is sharing a daily riddle designed to get your grey matter firing in the lead up to the festive season.

Do you think maths is fun? Well, his latest riddle is going to drive you around the Pentagon.

QUESTION

This pentagon has 10 circles on it: 5 at the corners and 5 in the middles of the edges. Place the numbers 1 to 10 in the circles so that each side adds up to 14. There is essentially only one solution. If you think you've found more than one solution they are just reflections or rotations of each other.

HINT

The numbers 1 to 10 have to appear in all the circles and each side must add to 14. So if we add up the 5 sides we will get 5 × 14 = 70. But if you look at the diagram, adding the sides will involve every circle once and the corners a second time. The circles contain the numbers 1 to 10 so adding the 10 circles up once gives us 1 + 2 + 3 + ... + 9 + 10 = 55. We need to get to 70 so when we add the corner circles a second time they must give us the extra 15. This tells you which numbers have to go in the corners. Play around with different combinations of these numbers in the corners and see how you go.

ANSWER

The five sides each add to 14, so adding all five sides together gives you 70. You can see from the diagram that adding all five sides together would count each circle around the pentagon but also pick up the five corners a second time.

Well all 10 circles around the pentagon must add up to 1 + 2 + 3 + … + 9 + 10 = 55. So the corners must give us the other 15 we need to hit 70.

If the five corners add up to 15, they must contain 1, 2, 3, 4 and 5.

Clearly the 10 can only sit in the middle of a side with corners 1 and 3. Similarly 6 must lie in between 4 and 5.

You can make further observations or just do trial and error from here to get a solution, starting from the top corner and reading clockwise of 1-10-3-6-5-7-2-8-4-9-1

If you liked that puzzle, try and find the only solution for a similar pentagon with sides adding to 19, the two different pentagons with sides adding to 17 and (really tough) convince yourself that there are no pentagons with sides adding to 16 or 18.