Eye of Horus

Slices that have to come to exactly one whole, and the question of how many different ways there are.

The Eye of Horus board

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Grades
3–6
Subjects
Fractions, Multiplying & Dividing, Logic
Kind
Puzzle
Original
Plastic Fraction Circles at MathPickle

What it is

A circle with its smallest slice already in it, say a twelfth, and a tray of slices: a half, a third, a quarter and on down. Fill the circle with slices the same size or bigger, so that they make exactly one whole. One rule: slices side by side may never add up to a single slice, so a third may not sit next to a sixth, because together they make a half. A row of empty circles shows how many ways there are, and the board is done when every one is filled. In Count the Slices the circle has to take an exact number of pieces as well.

The rules

Eleven keys, one after another: a circle with a half cut out of it in red and 1/2 written under it, then 1/3, 1/4 and so on, each slice thinner and in its own colour, down to 1/12.1/21/31/41/51/61/71/81/91/101/111/12
All the fractions in this puzzle will be Egyptian fractions. They have a 1 on top.
A circle with a twelfth already in it, outlined, just before twelve o'clock. An eighth goes in at twelve, then a quarter, a sixth and another quarter, each beside the last. An eighth of the circle is left empty.1/121/1211/121/81/121/1219/241/81/41/121/1213/241/81/41/61/121/123/81/81/41/61/41/121/121/8
In a puzzle, you’ll start with the smallest slice. Fill the circle with equal or bigger slices.
A full circle of quarters and twelfths, one after the other. A quarter and the twelfth beside it are outlined together, in the shape of the third they make, and everything else steps back. Then a third lies where they were, and a red cross is drawn over the circle.1/41/121/41/121/41/121/121/41/121/41/121/41/121/31/41/121/41/12
Make sure that no set of consecutive slices can be replaced by a bigger slice.

What it gives you to practise

Adding fractions with different denominators, and seeing the common denominator before writing it down: a quarter and a twelfth make a third because 3/12 and 1/12 make 4/12. Every slice has a 1 on top, the Egyptian fractions the puzzle is named for, so every check is whether some neighbours make one of those again. A few facts turn up on their own. A half never goes in, because the rest of the circle would be the other half, all in one run. With sevenths or elevenths as the smallest slice there is only one way: all sevenths, or all elevenths. With sixths there is one too: a third, two quarters and a sixth, with the quarters kept apart.

In a classroom

This is Gordon Hamilton’s Plastic Fraction Circles from MathPickle, which starts from a box of plastic fraction pieces and asks how many ways a circle can be built when the smallest piece is 1/6. The rule about neighbours is his as well, and the number of ways for each smallest slice is entry A394737 in the OEIS: 1, 1, 2, 5, 11 and 1 for sixths, sevenths, eighths, ninths, tenths and elevenths. A real set of fraction circles on the desk is a good companion, and a pair can check each other’s circles by hunting for neighbours that make one piece. Printed, a board is its circle with the smallest slice in it, and the rule in words.

Share or embed

The whole puzzle at one address, with every board in it. Whoever opens it stays inside it, and their progress stays in their own browser.

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