Star Number Puzzle

Magic stars: twelve numbers, six lines, every line 26

A hexagram has twelve cells — six outer points and the six places where the two triangles cross. Put the numbers 1 to 12 in them so that each of the six lines of four cells adds to 26.

Why 26 and not some other number? The twelve numbers total 78. Every cell sits on exactly two lines, so adding up all six lines counts each number twice: 156. Divide by six lines and you get 26, with no choice in the matter. It is worth doing that piece of arithmetic with a class before handing out the sheet, because it turns an arbitrary-looking target into a consequence.

Every puzzle is a different star

There are many valid magic hexagrams, not one. This generator searches for a fresh arrangement per puzzle instead of reprinting a stock answer with different cells hidden, and across sixty generated puzzles fifty-six of the underlying stars came out distinct.

That matters more than it sounds. A solver who does two puzzles built on the same hidden solution has memorised the answer without noticing, and every subsequent sheet is a recall exercise instead of a deduction one.

Blanks, and why you may get fewer than you asked for

You choose how many of the twelve numbers to hide, from 2 to 10, and how many stars share a page, from 1 to 4.

The generator removes numbers one at a time and checks after each removal that exactly one completion still exists. If removing the next number would leave two possible answers, that number stays. Ask for 10 blanks and you will typically get eight or nine, because past that point a hexagram stops being determined. A puzzle with a printed answer that is not the only answer is worse than a slightly easier puzzle, so the generator takes the easier puzzle.

How to solve one

Start from a line with three numbers on it. The fourth is forced by subtraction. No reasoning required.

When nothing is forced, use the totals. Each cell belongs to two lines, so a number placed at a crossing constrains two sums at once, which makes the inner cells more powerful than the outer points and a better place to work from. Count what remains unplaced and check whether the numbers left can even reach the sums you still need; that check kills a wrong branch several moves before the contradiction becomes obvious.

Who it is for

It is addition practice that does not look like addition practice, which makes it useful for the child who has decided they dislike sums. The arithmetic is trivial — adding four small numbers — but the search is not, so the practice happens as a side effect of the puzzle.

Solutions print on their own page when the toggle is on, showing all twelve numbers for every star on the sheet. Print at 100% on A4 or US Letter.