/ \ / A \ -----/-----\----- \ B / \ D / \ F / \ / C \ / E \ / X-----X-----X / \ H / \ J / \ / G \ / I \ / K \ -----\-----/----- \ L / \ / |
Draw a regular hexagon divided into six equilateral triangles.
On each side, place another equilateral triangle, to create a
six-pointed star.
Can you place the numbers 1-12 into the 12 triangles such that the sum of the five triangles in each of the six lines is the same? (For example, in the figure, BCDEF and GHIJK are two lines of five.) If it is possible, what are the largest and smallest common sums possible? If it is not possible, how many numbers would you have to change, keeping all 12 unique, to achieve the smallest possible common sum? |
Source: Original.
Sum 32; true values 1..12: / \ / 2 \ -----/-----\----- \ 3 / \ 8 / \ 5 / \ / 6 \ / 10\ / X-----X-----X / \ 7 / \ 1 / \ / 9 \ / 4 \ / 11\ -----\-----/----- \ 12/ \ /Another solution with a sum of 33 can be obtained by replacing each element with the same number subtracted from 13.
Sum=30; value 12 replaced by 13: / \ / 6 \ -----/-----\----- \ 8 / \ 7 / \ 10/ \ / 3 \ / 2 \ / X-----X-----X / \ 5 / \ 4 / \ / 9 \ / 1 \ / 11\ -----\-----/----- \ 13/ \ / Sum=31; value 11 replaced by 13: / \ / 5 \ -----/-----\----- \ 6 / \ 9 / \ 10/ \ / 2 \ / 4 \ / X-----X-----X / \ 7 / \ 1 / \ / 8 \ / 3 \ / 12\ -----\-----/----- \ 13/ \ / Sum=32; value 9 replaced by 13: / \ / 5 \ -----/-----\----- \ 6 / \ 11/ \ 8 / \ / 4 \ / 3 \ / X-----X-----X / \ 2 / \ 1 / \ / 10\ / 7 \ / 12\ -----\-----/----- \ 13/ \ / Sum=32; value 9 replaced by 14: / \ / 5 \ -----/-----\----- \ 7 / \ 10/ \ 11/ \ / 3 \ / 1 \ / X-----X-----X / \ 6 / \ 4 / \ / 8 \ / 2 \ / 12\ -----\-----/----- \ 14/ \ / Sum=32; value 11 replaced by 14: / \ / 3 \ -----/-----\----- \ 5 / \ 10/ \ 9 / \ / 7 \ / 1 \ / X-----X-----X / \ 4 / \ 6 / \ / 8 \ / 2 \ / 12\ -----\-----/----- \ 14/ \ / Sum=32; value 12 replaced by 14: / \ / 2 \ -----/-----\----- \ 5 / \ 8 / \ 6 / \ / 9 \ / 4 \ / X-----X-----X / \ 3 / \ 7 / \ / 10\ / 1 \ / 11\ -----\-----/----- \ 14/ \ /