Puzzle 1 of 26 · Hard

Labelling a die

Your answer

You write the numbers 1 to 6 on the faces of a blank cube, one number per face. Two labellings count as the same if you can turn one cube to look exactly like the other. How many different dice can you make?

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Show the solution

Answer: 30

Bottom

\displaystyle 1

Top

\displaystyle 5

Sides

\displaystyle 6

\displaystyle 30

Step 1 · Turn the cube so 1 faces down
$\displaystyle 1 \text{ way},\, \text{ once turning is allowed } = 1$
Any die can be turned so 1 is on the bottom. That uses up the freedom to tip the cube over.
Step 2 · Choose the number on top
$\displaystyle \text{any of the other } 5 = 5$
The top face is opposite 1, and turning can’t change which number is opposite 1.
Step 3 · Arrange the 4 sides in a ring
$\displaystyle (4 - 1)! = 3! = 6$
The 4 side faces form a ring you can still spin, like a round table. Fix one number and arrange the other 3: 3! ways.
Step 4 · Multiply
$\displaystyle 5 \times 6 = 30$
Each choice of top goes with each ring order, so multiply.

Answer $\displaystyle 30$

Step 4 of 4

The method · Unit 1, CountingPermutationsCount arrangements when order matters

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