An easy one for the middle of the week!
Jane and John were discussing a business idea. John wanted to make a little set to keep track of the date. It was to include two cubes with a single digit on each face that could be rearranged to show any day of the month (1-31).
"Look," he said, "there are 72 ways of arranging the cubes, so it must be possible to design it for just 31 days!"
"No," responded Jane, "there's less than 72. Presumably each cube has 6 different faces, so at least 2 are going to be overlapping. That's going to reduce it to... mmm"
[Question 1: how many different ways are there to put down the cubes to read out a two digit number in the condition that Jane just described?]
"Well, that should still be plenty." John replied.
"Hang on, there's another problem." Said Jane. "You have got 10 digits altogether. You will need 0, 1, and 2 on both cubes, since you need to make 11, 22, and 0n for all n between 1 and 9 inclusive. That means you need to pair 0 with at least 9 other numbers. So it's got to be on both. Amiright?"
"I guess so. The argument for 0 applies to 1 and 2 also. Either way, I agree. Both cubes are going to need 0, 1, and 2"
"That leaves 3 spaces on each cube for the other numbers, or a total of 6 spaces. But there are 7 numbers left: 3 to 9." Jane continued.
"I see what you mean. Maybe I should use dodecahedrons. I prefer them anyway." Said John thoughtfully.
"Wait!" Said Jane. "Here's one for sale on the internet. How do they do it?"
[Question 2: How do they do it?]