The hexagonal pattern of circles illustrated below contains the numbers 1 to 13.

There are in fact nine lines each containing three circles. The numbers along five of these lines (2, 9, 11; 7, 2, 13; 8, 9, 5; 3, 9, 10; 13, 3, 6) sum to give the same total of 22.

Is it possible to rearrange the 13 numbers in such a way that all nine lines have a common total? What would this total be? And what number (or numbers) is allowed in the centre if this task can be completed?

Hint 1

Can we use the obvious similarities to the magic square problem (Aha! 2)? Perhaps a similar plan of attack would work here?

Hint 2

What is the sum of all the numbers? What is the total if we add all the rows together? Why is the central number special’?

Solution

In the magic square problem we added together the numbers in all the rows and divided by the number of rows to obtain the common row sum. We try this strategy here, but it is more complicated because each number is counted more than once. The Aha! here is the realization that each number other than the one in the centre lies on exactly two rows, and is therefore counted twice in such a sum. The central number, C, will be counted three times. The total sum then will be 2 x 91 + C = 182 + C, where C is 1, 2, 3, ... or 13. Remembering that 182 + C (hopefully) gives the sum of nine rows having common sum, 182 + C must be exactly divisible by 9. We deduce that if there is a solution, then C = 7, and the common row sum is 21.

With a little trial and error, a possible solution now is to have 7 in the centre, and around the outside, starting from a vertex,  8, 1, 12, 5, 4, 11, 6, 13, 2, 9, 10, 3.

Extensions

1. In the magic hexagon, what happens if you subtract 1 from each number? Do we still get a magic hexagon? Can you suggest how to place the numbers 0, ±1, ... , ±6 in a magic hexagon? Extend this idea.

2. How would you define a 3 x 3 x 3 magic cube? If the numbers used were 1, 2, 3, ... , 27, what would the central number be? What would a ‘face sum’ be?

3. See if you can place the numbers 1 to 9 in the Pascal configuration below to make a magic figure.

Use your library to find other such configurations for example, the Desargues configuration.

Hint 1

Hint 2

Solution

Extensions