John Conway

Around 25 years ago I was at a conference at Princeton, in whatever building housed the math department at the time. One of the sessions looked like it would be kinda boring, so I took a stroll down the hallway and came to a lounge, a cozy little place of the sort that you’ll see in an out-of-the-way corner of a university, with a couple of couches and tables and a shelf with a mix of old books and recent journals. The room was empty except for a couple of students working quietly in a far corner and a heavyset bearded middle-aged man playing with some blocks. I took a seat nearby and he explained what he was doing.

He had 27 identical wooden blocks—they weren’t cubes, I guess you’d call them rectangular parallelepipeds? I didn’t need to know the word for it because I could see the blocks in his hand—along with a wooden box that could hold all them, if they were fit in just right. If the boxes had dimensions a x b x c, then the box had dimensions (a + b + c)^3. (Again, this didn’t need to be explained, because the pieces were right there in front of us.) The blocks can just fit into the box in some grid (e.g., (((a, b, c), (b, c, a), (c, a, b)), etc.)), but it’s not just a combinatorial challenge (I’d say a Sudoko-like challenge but this was before I’d heard of that particular puzzle) but also a geometrical challenge, because if you just try to cram the pieces in the box any which way, they’ll interfere with each other. It’s also a cool puzzle because (a) if you can fit the pieces in, there will be room to spare, some empty space in the interstices, and (b) all the 27 pieces are identical, and how cool is that?

The bearded heavyset man did not actually say, “how cool is that?” Instead, he pointed out to me in his English accent that the problem is challenging, and by contrast the two-dimensional version (with 4 identical rectangles) is trivial (as indeed it is, as you can see from a moment’s reflection). He also claimed that the four-dimensional version isn’t hard to solve—somehow you just put together two 2-dimensional solutions—and he said that he didn’t know if the five-dimensional version had a solution. I thanked him and left the room. We never had any introductions; he just jumped in and showed me the puzzle, that was it.

A few minutes later I was running this episode through my mind . . . Princeton mathematician . . . English accent . . . puzzles . . . it must have been John Conway! But I didn’t try to track him down or ask for his autograph or whatever. Why ruin the perfect moment?

Later I was visiting my parents—my dad had a workshop in the basement and I decided to make a version of the puzzle for myself. In the version Conway showed me, each block was about the size of my hand, but to reduce the effort I decided to make something smaller. The only constraint in making these a x b x c blocks is that, if a a + b + c. Otherwise it’s possible to cheat and squeeze the blocks in four at time. I chose a, b, c to be in the proportions 4, 5, 6. So I got a big board, sawed it into pieces, sanded the to be just right, and stained them. For the box I got some pieces of plastic and taped them together at their edges.

Then I sat down to solve the puzzle. It took me a couple hours! Indeed it’s not so easy. I guess I could’ve done it quicker by taking some notes while I was doing it and working through by process of elimination. In any case, when I finally did solve it, it was satisfying.

I still have that wooden puzzle. It’s in my office. I doubt Conway invented it. But I thank him for showing it to me. Also I thank him for coming up with the game of life. Talk about cool.