# Caesar’s Calendar puzzle

So I got this Caesar’s Calendar puzzle as an early Fathers’ Day gift. (I will often get various puzzles as gifts.) Link to mine but others seem to produce it. (This page also has a link to solutions but I don’t want to click on it yet.)

It … is … amazing.

It’s sort of a perpetual calendar thing with rectilinear block pieces arranged to a grid. You are to place the blocks so that there are 3 open squares. Each square exposing one of the month name, day of the month and day of the week.

More deets: The grid is basically 7x8 square. There are 12+31+7=50 needed squares. The extras are incorporated into the surrounding frame. The base text is laid out as follows (x’s for the extras):
Jan - Jun X
Jul - Dec X
1-7
8-14
15-21
22-28
29-31 Sun - Wed
X X X X Thu - Sat

The blocks are the usual assortment of shapes made of 4-5 sub-squares: L’s zig-zags, etc. The T, U and straight-4 seem to be particularly interesting to deal with. The front and backs seem similar in shading, I think flipping the pieces over is needed sometimes.

I see many people describing this puzzle as having 365 solutions when:
a. Clearly the number should be 366 for leap years.
b. And 7 times that for days of the week.

I am assuming that while you might be able to solve for Sun Feb 31, that doesn’t “count”.

I have played with it a little finding that just putting all the pieces back in regardless of the exposed squares to take a few minutes. The YouTube videos demonstrating particular solutions indicate that it does take a bit to solve.

Now, multiply that by the 1862 different solutions you will need to solve for over time and …

But, that’s not why I’m here. I keep thinking about what it took to come up with this is the first place. There has to be a system underneath it all. Do I still have what it takes to figure out the system? (And do I still have the time?)

Anyway, I got this last night and the grandkid promptly jumped in to play with it, saying “I like puzzles.” Once we got the pieces back in, in any old order, the cry of “Again!” sounded out. Not bad for a 3 1/2 year old. The nerdness is strong in this one.

In the 2nd solution video above there’s also a review of another perpetual calendar puzzle but based on Rubik’s cube. Fun fact: This puzzle works out well in English since the cube has 6 sides are there are only 6 different letters in the middle of the month abbreviations. Jun-Jul-Aug really helps.

I’ve always liked puzzles like this. I had a Soma and pentominoes when I was a kid. The book that came with the Soma included some analysis that was over my head at the time, but I can appreciate now.

A friend of mine in college had a calendar made of cubes, like this:

You need to be able to make the numbers from 01 to 31 with two cubes. One of them has 0 1 2 3 4 5, and the other has 0 1 2 6 7 8. It only works because you can turn the 6 over to become a 9. A quirk of typography makes it possible. And there are 12 months, so you can put 2 on each face of a cube.

I would (double-)check the design using a computer.

That Rubik’s cube thing I mentioned has got me thinking.

My favorite Platonic solid is the dodecahedron. We had one that was a calendar when I was a kid. But of course it was only for one year. I now have a dodecahedron where I print out and stick on the months for the upcoming year each January.

There’s a dodecahedron version of Rubik’s cube. Can you make an annual calendar design based on it that gives you a perpetual calendar? Lots of tricky stuff plus you might have to deal with all months showing 31 days. And maybe the left most column in a month might not be Sunday.

Do you mean this?

What do you mean by all months showing 31 days? I assume the idea is to rotate only the number(s) you want to the current month’s face, orient them correctly, and hide the others on other faces?

I was fiddling with puzzle earlier today and it didn’t take too long before I got my first solution. I was so happy. Was ready to take a pic of it. While looking it over I realized it had:

Jun 15 Fri.

What?

The really silly part is that I actually started filling in around the Fri. Sad face.

Re: A calendar dodecahedron. In a perfect world, each face would have one month of the year.

Lotta these

but I would not call it much of a puzzle…

I imagine that the original puzzle was based on picking any three random squares in the grid to exclude, and trying to find an arrangement of the pieces that works. What’s new with this one is just putting labels on those squares and calling it a calendar. So not only would it be possible to solve for Tuesday the 31 of February, but it would also be possible to solve for 17 the Monday of 12, or July the Sunday of October.

Maybe the designers discovered, via exhaustive computer search, that there were a few arrangements of open squares that led to an impossible puzzle, and contrived to arrange the labels in such a way that all of those were invalid dates.

Given the orderliness of the text they didn’t seem to do anything special to arrange those. The real option is the location of the blocked out squares near the edges. These seem to make the puzzle harder. The lower right area is sort of hemmed in and gives me the most trouble.

Anywaaay. I finally solved my first current date last night. For reals this time. I didn’t just adjust the earlier mistaken solution. I dumped all the pieces out and started over.

I think the key seems to start in that lower right corner where the days of the week are.

I have come to think that there is no esp. helpful system for solving it. I had considered “arity” , possibly a mod 3, thing. So an exhaustive computer search to verify all actual dates were achievable seems likely to have been done by the creator.

Note that there’s a simpler version without the days of the week. So somebody may got the idea to extend it, went back and forth with selecting which blocks to include and computer testing.

Another date solved last night. (I do puzzles while watching TV in the evening, FWIW.)

So I think I’m getting the hang of it. I’ll report back after I’ve gone thru all 1862 dates.

Re: static dodecahedron calendar. I download a printout/cutout for the new year, scale it, print it, and cut out the individual pentagons and paste onto my (heavy) dodecahedron. I like some heft to my desk fidget things.

I don’t know where this number came from, it should be 2562, that is 366*7 as noted.

Like so much, it’s taking longer than we think.

A few years ago I wrote a program to solve tile puzzles like this one. You enter the list of tiles and the board shape, and it produces all solutions. It translates the problem into an exact cover problem and then uses Knuth’s Algorithm X to solve it. I tried it on the example that’s shown in the photo (Sunday Jan 1). I started it running yesterday and it’s still running, having found 686,791 solutions so far. So apparently this set of tiles is pretty adaptable. I suspect having the three rectangular tiles contributes to that. Here are a few of the solutions my program found.

``````    +-----------+---+-------+
|           |   |       |
+---+---+       |   +---+   |
|       |       |   |   |   |
+---+   +-------+   |   |   |
|   |           |   |   |
+---+   +-------+---+   +---+
|   |   |       |           |
|   |   +---+   +---+---+---+
|   |   |   |       |   |   |
|   +---+   +-------+   |   |
|       |           |   |   |
+---+   +---+---+   |   |   |
|   |   |   |   |   |   |   |
|   +---+   +---+---+   |   |
|           |       |   |   |
+-----------+-------+---+---+

+-----------+-----------+
|           |           |
+---+---+       +---+   +---+
|       |       |   |   |   |
+---+   +-------+   |   |   |
|   |           |   |   |
+---+   +-------+---+---+   |
|   |   |       |           |
|   |   +---+   +---+---+---+
|   |   |   |       |   |   |
|   +---+   +-------+   |   |
|       |           |   |   |
+---+   +---+---+   |   |   |
|   |   |   |   |   |   |   |
|   +---+   +---+---+   |   |
|           |       |   |   |
+-----------+-------+---+---+

+-----------+-----------+
|           |           |
+---+---+       |   +-------+
|       |       |   |       |
+---+   +---+---+   +---+   |
|   |   |   |   |   |   |
+---+   |   |   +---+   |   |
|   |   |   |   |   |   |   |
|   |   |   |   |   +---+---+
|   |   |   |   |           |
|   +---+   |   +---+---+   |
|       |   |   |   |   |   |
+---+   +---+---+   |   +---+
|   |   |   |   |   |       |
|   +---+   +---+   +---+   |
|           |           |   |
+-----------+-----------+---+

+-----------+-----------+
|           |           |
+---+---+       |   +-------+
|       |       |   |       |
+---+   +-------+---+---+---+
|   |               |   |
+---+   +-----------+---+   |
|   |   |           |   |   |
|   |   |   +-------+   |   |
|   |   |   |           |   |
|   +---+   |   +---+---+   |
|       |   |   |   |   |   |
+---+   +---+---+   |   +---+
|   |   |   |   |   |       |
|   +---+   +---+   +---+   |
|           |           |   |
+-----------+-----------+---+

+-----------+-------+---+
|           |       |   |
+---+---+       |   +---+   |
|       |       |   |   |   |
+---+   +---+---+   |   +---+
|   |   |       |   |   |
+---+   |   +-------+   |   |
|   |   |   |           |   |
|   |   |   +-----------+   |
|   |   |   |           |   |
|   +---+   |   +---+---+   |
|       |   |   |   |   |   |
+---+   +---+---+   |   +---+
|   |   |   |   |   |       |
|   +---+   +---+   +---+   |
|           |           |   |
+-----------+-----------+---+

+-----------+---+-------+
|           |   |       |
+---+---+       |   |   +---+
|       |       |   |   |   |
+---+   +---+---+---+   |   |
|   |   |   |       |   |
+---+   |   |   +-------+   |
|   |   |   |   |           |
|   |   |   |   +-----------+
|   |   |   |   |           |
|   +---+   |   +---+---+   |
|       |   |   |   |   |   |
+---+   +---+---+   |   +---+
|   |   |   |   |   |       |
|   +---+   +---+   +---+   |
|           |           |   |
+-----------+-----------+---+
``````

The program finally finished. It found 744,182 solutions.

A recap of my recent (human) solutions. I did 4 dates in a row, each taking a bit of time. Failed on one date (I hadn’t slept well the night before) but then things turned around. I did two dates rather quickly with virtually no trial and error. One was probably under a minute. Then another longer time and then a couple more fast ones.

Last night I did yesterday’s date as fast as I could put tiles down with no backups.

One thing that appears to be happening is that the subconscious part of my brain has gotten good at this and a lot of what I’m doing consciously is merely channeling the moves without really overtly thinking.

So, it’s a lot easier than I first thought (for the most part), there could several solutions per date, etc.

I don’t understand a lot of what’s going on with markn_1’s post. There are not 3 rectangular pieces. There is only one (the long one). You may have confused the 2 fixed indentations of the board with movable tiles. Note the X’s in the layout I gave in the OP.

(I quite understand exact cover, etc. I’ve published several papers on completeness results for various complexity classes.)

Oh, you’re right. Since the packaging covers up the edge of the board, I couldn’t tell that those are part of the board and not separate tiles. The 1x2 in the top right and the 4x1 in the bottom left are part of the board, not tiles as I thought, right?

That actually makes the problem much faster to solve. There are 1495 solutions to the Sun Jan 1 puzzle, which my program found in 1.07 seconds.

When I see combinatorial puzzles like this, I tend not to think, “how can I solve this?” but rather “how can I write a program to solve this?” It kind of bugs my wife when she buys me a puzzle and instead of working on it, I write a program to solve it.

The first 8 of the 1495 solutions:

``````    +-----------+-------+
|           |       |
+---+---+---+   +---+   |
|       |   |       |   |
+---+   |   +---+---+   +---+
|   |       |       |   |
+---+   +---+   +-------+   |
|   |   |   |   |           |
|   |   |   +---+---+---+   |
|   |   |           |   |   |
|   +---+-------+   |   +---+
|   |           |   |   |   |
|   |       +---+---+   |   |
|   |       |   |       |   |
+---+-------+   +-------+   |
|           |
+-----------+

+-----------+-------+
|           |       |
+---+---+---+   +---+   |
|       |   |       |   |
+---+   |   +-------+   +---+
|   |           |       |
+---+   +-------+---+-------+
|   |   |       |           |
|   |   +---+   +---+---+   |
|   |   |   |       |   |   |
|   +---+   +-------+   |   |
|   |       |           |   |
|   |       +---+---+   +---+
|   |       |   |   |   |   |
+---+-------+   |   +---+   |
|           |
+-----------+

+-----------+-------+
|           |       |
+---+---+   +---+---+   |
|       |   |       |   |
+---+   |   |   +---+   +---+
|   |   |   |       |   |
+---+   +---+   +---+---+   |
|   |   |       |   |       |
|   |   +---+---+   |   +---+
|   |   |   |       |   |   |
|   +---+   |   +---+   |   |
|   |       |   |   |   |   |
|   |       +---+   +---+   |
|   |       |   |   |       |
+---+-------+   |   +-------+
|           |
+-----------+

+-----------+-------+
|           |       |
+---+---+   +---+---+   |
|       |   |   |   |   |
+---+   |   |   |   |   +---+
|   |   |   |   |       |
+---+   +---+   |   +-------+
|   |   |       |           |
|   |   |   +---+---+-------+
|   |   |   |       |       |
|   +---+---+---+   +---+   |
|   |           |       |   |
|   |       +---+---+---+   |
|   |       |   |   |       |
+---+-------+   |   +-------+
|           |
+-----------+

+-----------+-------+
|           |       |
+---+---+   +---+   +---+
|       |   |       |   |
+---+   |   +-------+   +---+
|   |   |           |   |
+---+   +---+   +---+---+   |
|   |   |   |   |   |       |
|   |   |   +---+   |   +---+
|   |   |           |   |   |
|   +---+-------+---+   |   |
|   |           |   |   |   |
|   |       +---+   +---+   |
|   |       |   |   |       |
+---+-------+   |   +-------+
|           |
+-----------+

+-------+-----------+
|       |           |
+---+---+   |   +---+   |
|       |   |   |   |   |
+---+   |   +---+   +---+---+
|   |       |       |   |
+---+   +-------+---+   |   |
|   |   |           |   |   |
|   |   +---+   +---+---+   |
|   |   |   |   |   |       |
|   +---+   |   |   |   +---+
|   |       |   |   |   |   |
|   |       +---+   +---+   |
|   |       |   |       |   |
+---+-------+   +-------+   |
|           |
+-----------+

+-------+-----------+
|       |           |
+---+---+   +-------+   |
|       |   |       |   |
+---+   |   +---+   +---+---+
|   |       |           |
+---+   +-------+-----------+
|   |   |       |           |
|   |   +---+   +---+---+   |
|   |   |   |       |   |   |
|   +---+   +-------+   |   |
|   |       |           |   |
|   |       +---+---+   +---+
|   |       |   |   |   |   |
+---+-------+   |   +---+   |
|           |
+-----------+

+-------+-------+---+
|       |       |   |
+---+---+   +---+   |   |
|       |       |   |   |
+---+   +---+---+   |   +---+
|   |   |       |       |
+---+   |   +-------+---+   |
|   |   |   |           |   |
|   |   |   +---+   +---+---+
|   |   |       |   |       |
|   +---+-------+   |   +---+
|   |           |   |   |   |
|   |       +---+---+   |   |
|   |       |   |       |   |
+---+-------+   +-------+   |
|           |
+-----------+
``````

Let’s see, even taking the 1.07 seconds to find all solutions for one date and multiplying that by 2562 dates gives about 2741 seconds or 45.7 minutes. It would be a lot less to test for just one solution for each date. With some variability given some dates may be easier or harder. (Having the open spots in the middle of the puzzle, unlike Jan 1, makes this harder.) It really does seem quite plausible that the designer just tried out various tile sets and tested them on a computer.