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It All Adds Up


Place the pool balls (numbered 1 to 9) in the circle so that they add up to the same number across each diameter line. How many different ways can you solve puzzle?

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There are eight depressions in the tray round the outside of the circle and one in the middle, so there are four diameters outlined in red across the circle. So the numbers 1 to 9, with one removed to go into the middle, have to be divided up into four pairs that add up to the same total.

Clue 1


An easier way to solve this puzzle is to find number pairs that add up to the same total, rather than trying to add three numbers together. The way this puzzle is designed means that every line will share the same central number, so focus on finding number pairs that give the same total, then double check your work by placing the spare number in the centre.


Clue 2


When looking for pairs of numbers that add to the same total, try adding one of the highest numbers to one of the lowest.  Then try getting three other pairs of numbers to add to that same total.


Clue 3


The three solutions to this puzzle have the lines adding up to 12, 15 and 18.

How can we be sure there are only three answers?

Each outer pair of numbers will add up to the same total. As there are four pairs of numbers, we know that the combined total of these outer number pairs must divide by four to give a solution. We can also see that the numbers 1 to 9 to add to 45 (a quick way to check this is to look at the middle answer above: four pairs adding to ten, plus the five gives us 45). So if we take one number away from 45, as it will be in the centre of the puzzle, what must the other eight numbers add up to?

Its in the division

With 1 in the middle, the other eight numbers add to 45 – 1 = 44, the highest possible total for the eight numbers.  This divides by four to give 11, and so we had four pairs that added to 11. With 5 in the middle, the rest of the numbers add to 45 – 5 = 40, which give four pairs that add to 10. With 9 in the middle, the other numbers add to 45 – 9 = 36, which divides by four to give nine, and is the lowest possible total for the numbers. There are no numbers in the range from 36 to 44 that divide by four with no remainder; so these are the only possible answers.

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Welcome to Look Closer, Techniquest's brand new digital tool. We're trialling Look Closer as a way of offering you a closer look at the science demonstrated by many of our exhibits. You'll find visitor sheets, videos, articles on contemporary science as well as fun trails to navigate your way around the exhibition space.

Speak to our Science Communicators to find out more.