Reasoning · Grade 5-2 Number Arrangements

Problem

Count distinct arrangements up to rotation

Four squares are joined into an S-shaped tile. Write 1, 2, 2 and 3 into the four squares, one each. Two fillings that match after a turn count as one. Count how many genuinely different fillings there are.
Your answer
How to solve
Strategy Make a Systematic List — There are only a handful of fillings, so the safest route is to list every one of them and then cross out the duplicates -- no formula needed and nothing left to chance. The one thing that has to be settled before listing is what a turn actually does to this figure, which is a spatial question: turning the S tile a quarter turn stands it on end, so it no longer covers its old squares, while a half turn puts it back exactly where it was but swaps the squares in pairs. Cutting the shape out of paper and turning it (tool 10) makes that fact obvious in a second and stops the usual mistake of dividing by 4 for four possible turns. Once the half turn is understood, the twelve fillings pair off and the count follows.
1STEP 1

Count the fillings before worrying about turning

Ignoring turns there are 12 fillings.

4 × 3 × 1 = 12
2STEP 2

See what a turn does to the S tile

The S tile returns to itself under a half turn.

3STEP 3

Check whether any filling is unchanged by the half turn

No filling is left unchanged by that turn.

4STEP 4

Pair the twelve fillings off

So the twelve fall into pairs.

12 ÷ 2 = 6
5STEP 5

Confirm the count a second way

That leaves 6 genuinely different fillings.

3 + 3 = 6
Answer
6 fillings
12 ÷ 2 = 6
The answer is a plain count of pictures, so it should be a whole number no bigger than the 12 fillings we started from and no smaller than 12 ÷ 4 = 3, which is what you would get if all four turns did something. 6 sits sensibly between those bounds, and it is exactly 12 ÷ 2 because only the half turn acts on this shape. The two independent methods -- pairing the twelve, and pinning the 1 into A or B -- both give 6. A common wrong answer is 12 (forgetting that turning is allowed) and another is 3 (pretending all four quarter turns match the figure onto itself); both are ruled out by actually turning a paper tile.
Takeaway

Cut the shape out and turn it first -- this staircase only matches itself after a half turn, so the twelve fillings pair up and just 6 are really different.

  • Count the fillings before worrying about turning
  • See what a turn does to the S tile
  • Check whether any filling is unchanged by the half turn
  • Pair the twelve fillings off
  • Confirm the count a second way