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Question

Dear Mitch, 

This probably isn't one of those letters you'll put up on your website, but after reading that question somebody wrote in about father's day and all that, I was kind of curious if you gave your own father some kind of math thing, or if you just got him a regular present like a tie. (Also, I know your hobby is building things out of recycled materials, and I was curious if you do that for gifts too.  I mean do you think people really like homemade stuff that is part cleaned up garbage?  Wouldn't most people rather get something new and clean from a store? 

Just curious, and hope I'm not being too personal, but I always wondered about that.

Yours truly, 

Clive W.

Omaha, NE

Answer

Dear Clive W.,

Oh, I very much want to post this letter on my site, and in a moment you'll see why.

As it turns out, I did give my father a gift that is related to math and problem-solving, and it was indeed one that I made out of recycled materials (terms I've come to prefer over 'garbage', the main distinction for me is that the materials were all completely clean, which is important for gift-making, even if the gift you are making is for yourself!) 

I made my Dad something that I have made three or four times in the past for other people (perhaps over a span of ten years or so);  I recall making one for my son, one for a woman with whom I worked for years in a school on the beautiful East End of Long Island, and the final one I recall making before my Dad's was for a neighbor in the Hamptons, as a token of appreciation for insisting I utilize a guest cabin on his property whenever I needed to finish a project that required more focus and fewer distractions than one could ever expect to encounter so conveniently!

I have recently learned that I am not the first to make this item, and they've been around for so long that there's no chance that I invented it, yet I seem to recall making them long before I knew they could actually be purchased. 

Okay, enough preamble; now, the problem-solving aspect to it:

Basically, the thing is a three-dimensional day-by day calendar consisting of three cubes.

Two of the cubes have a single number printed on each of its six sides, and the third cube has a day of the week printed on each of its six sides.

So far, there doesn't seem to be a problem to solve, right?

Using the two number blocks, the recipient, who has usually placed it on his desk or bookshelf, begins his/her day by rotating the blocks so that the day of the month is represented.  For example, on the 21st of the month, the person would turn the block on the left to show a 2 and the second block (the one on the right-hand-side) to show a 1.  Next, of course, the person turns the day-of-the-week cube to show what day it is, such as Monday, Tuesday, or Wednesday, etc. 

The first question that most people ask after having a moment to ponder the three cubes is this: Since a cube only has six sides, and there is only one number printed on each side, and our number system uses ten digits, how can all two-digit numbers be created?

Most people respond to their own question within seconds, saying, "of course, it's obvious, since the greatest number of days any month has is thirty-one, you only need three digits on one of the blocks – 1, 2, and 3.  And that sounds pretty good, but after fiddling with the blocks for another moment they realize there is more to the story. 

Why? 

Because: First, you need two 1's to form the 11th of each month, you need two 2's to form the 22nd of each month, and since both blocks are to be used even when the date is only a single digit, you need a zero printed on each block to be able to create 01, 02, 03, 04, 05, 06, 07, 08, and 09.  Otherwise, with only a single zero on one cube, you would need a 1, a 2, a 3, a 4, a 5, a 6, a 7, an 8, and a 9 on the other cube, which of course would require that cube to have more than six sides. 

So... between the 12 sides of the two cubes, you need this collection: 1, 1, 2, 2, 3, 0, 0, 4, 5, 6, 7. 8, and 9. (Don't forget about the 14th of the month, the 15th, 16th, 17th, 18th, 19th, 22nd, and the11th.  

And that's the problem:  you need 13 sides to do it, yet the two cubes only provide 12 sides...

How can this quandary be resolved?

Stay tuned....

Because if you haven't figured it out by tomorrow (or the following day), I will reveal the simple solution.

And as to the seven days of the week being printed on a six sided cube, well, that's too silly to waste any more of your time now.  That too I will reveal with the answer to the digit question above.  I'll also tell you a quick way to make this fun gift; either people really seem to like it or they are very effective at saying whatever is necessary to avoid hurting my feelings.  And that, I would guess is about a 50-50 chance.  In any case, the blocks seem to be on people's desks much longer than I would ever predict!

Until Next Time,

Mitch