## 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 < b < c, it's necessary that 4a > 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.

P.S. Terence Tao reports that “Conway was fond of hanging out in the Princeton graduate lounge at the time of my [Tao’s] studies there, often tinkering with some game or device, and often enlisting any nearby graduate students to assist him with some experiment or other.” So maybe Tao was one of those students quietly working in that far corner. Some good stories in the comments to that post too.

P.P.S. Reading Tao’s post more carefully, I notice this

He [Conway] challenged me to a board game he recently invented . . . I still remember being repeatedly obliterated in that game, which was a healthy and needed lesson in humility for me (and several of my fellow graduate students) at the time.

This is a funny story for a couple of reasons. First, Conway invented the damn game; it should be no surprise he could beat you at it, right? Second, when my friends and I were in grad school, we were in awe of the faculty. No lessons in humility were needed. So it’s funny for me to think that these grad students needed to lose a board game in order to get that feeling that we had all along. I guess this is related to the point made by Dick DeVeaux that math is like music, statistics is like literature. Math students come in with the ability to do everything, but statistics students are aware that there’s a world of things to learn.

1. Terry says:

So the puzzle is generalizable to any a, b, and c (with some mild restrictions)?

If so, does the solution always follow the same pattern? (If it doesn’t that would be really wild.)

• Andrew says:

Terry: As long as a < b < c and 4a > a + b + c, I don’t think the solution should depend on the specific values.

2. Carlos Ungil says:

This is Hoffman’s packing problem: https://en.wikipedia.org/wiki/Hoffman%27s_packing_puzzle

There is a variant from Knuth where the constraint is 4a = a + b + c (v.g. 3 x 4 x 5) and the challenge is to fit 28 blocks in the box.

• Andrew says:

Carlos:

Thanks for the link. It’s good to know where the puzzle came from. Its inventor deserves credit for coming up with something so beautiful!

3. Carlos Ungil says:

A nice memorial: https://xkcd.com/2293/

4. More Anonymous says:

The board game mentioned in the story above is a lot of fun and I recommend it, especially if you are stuck home from work and don’t mind getting addicted enough to spend hours with it.

It’s called “Phutball,” or “Philosophers’ Football,” or “ConwayGo.”

• Bill Spight says:

Phutball is the only game I ever saw Conway play. Back in the 1970s, I have heard, Conway would often invent 10 games a day.

I met him at a conference in 1994, two middle aged hippies at the back of the room. ;)

5. David J. Littleboy says:

IBM donated an IBM 1130 to my (males only at the time, it went coed the year after I graduated) high school. One of the first programs I wrote (this was 1970) was, of course, Conway’s Life. I had the line printer printing out life patterns, and it was making a decent rhythm, so a friend and I linked arms and danced to it. The math teacher who was supposed to be teaching us programming opened the door, looked in, closed the door, and was never seen in the basement computer room again.

I’ll second the thank you John Conway bit.

6. Bill Jefferys says:

I taught a course on “Time” many times at the University of Texas…everything from how the ancients kept track of the seasons to general relativity. The course was for non-science students who had taken our introductory astronomy course.

One of the things I covered was Conway’s elegant “Doomsday” method for determining the day of the week for any date in history, in your head. I find it very handy and use it quite frequently myself, for example, to write dates on a check (knowing what day of the week it is…the inverse problem). My web page for this uses a simpler method for calculating the “Doomsday” for a given year than the one Conway originally came up with, but here it is:

I’ll also mention (this has nothing to do with Conway) that the section on relativity used a wonderful book by Lewis Carroll Epstein, “Relativity Visualized”, that in a mathematically accurate way makes the concepts in relativity completely transparent and understandable using diagrams drawn in Euclidean space. Nothing more complex than understanding Pythagoras’ theorem has to be assumed, and anyone coming to the university would have seen this in high school geometry/algebra classes. Epstein used a very clever trick to turn the usual hyperbolic metric of special relativity:

dđťť‰^2=dt^2-dx^2

into a Euclidean metric:

dt^2=dđťť‰^2+dx^2

He then exploits this to get all the standard results of both special and general relativity. The charts at my website are the ones I used in class (although I deliberately left out some when I was asking students to guess what happens next…)