# Problem: The Desk Calendar

I love this little puzzle.

Some years ago (I was living in Canberra at the time), my wife (well, she wasn’t my wife yet) and I were at a friend’s place and I happened to notice a gorgeous little calendar on their kitchen windowsill.

According to this calendar, it was the 25th of April. My friend told me that each morning she would use the two blocks to write out the day of any month. This means that just using those two blocks, she can make all of the numbers from 1 to 31:

$01, 02, 03, \ldots, 29, 30, 31$

I didn’t think much of it at the time, but when I got home later that night, the little cube calendar found its way into my mind and my brain started to do its thing. I was trying to work out what the numbers on the unseen faces of the cubes were. In fact, that’s the problem I’m going to give you. Enjoy!

Problem. Look at the picture of the desk calendar. What are the four numbers that cannot be seen on the left cube and the three numbers that cannot be seen on the right cube?

Hint: If you haven’t solved this after some time then try gluing your feet to the ceiling.

1. Thomas Morrill says:

Very neat puzzle! I think the parameters will constrain us to certain fonts for writing numerals.

Start with the right cube in the picture. Both the dates 11 and 22 must be representable, so the sides of this cube are labelled 1, 2, 3, 4, 5, with the sixth side to be determined. At this point there are five unknown slots for digits: four sides on the left cube, and one on the right. There are also five digits that have not been placed, 6, 7, 8, 9, and 0.

It appears we just need to arrange the missing digits on the unknown sides, but there is a hitch. The dates 01, 02, 03, 04, 05, 06, 07, 08, and 09 must all be representable. In other words, there must be a 0 on the cube opposite from each of the digits 6, 7, 8, and 9.

Easy enough, put the 0 on the right cube. But then 03 is not representable! Okay, put 0 on the left cube, say with 6, 7, and 8. Then 06, 07, and 08 are not representable. Dear.

This looks to be impossible, but only as long as we assume we are trying to allocate five unique digits to five slots. The trick is to write the digits 6 and 9 as rotations of one another, so that they both occupy a single slot, and put a 0 on both cubes.

The left cube reads 0, 1, 2, 6/9, 7, 8, and the right cube reads 0, 1, 2, 3, 4, 5.

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• Yes it seems a little bit of combinatorics can be a hindrance, not a help here, haha…

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