This year, I’m writing a book about mathematical games. For my research, I’ve been reading old collections, scrounging Board Game Geek, hosting play-test parties, attending game design conventions, and conscripting students as guinea pigs (which works much better than the reverse: conscripting guinea pigs as students).

Nowhere in all my research have I come across a mind quite like that of Walter Joris.

Walter generates games, puzzles, and pencil-and-paper experiments with such intensity and regularity that he must be a kind of pulsar: some heretofore unknown astronomical object, emitting what I admiringly call Joris Radiation.

You and I see right angles. Walter sees a doodle game.

You and I see paper. Walter sees the Incredible Paperman.

You and I see a cube. Walter sees… well, to be honest, I don’t know *what* Walter sees, but I can’t help wanting to see it too.

His book *100 Strategic Games for Pen and Paper* is the most bizarre and marvelous thing I’ve read this year. “Nearly all the games have been invented by me,” he writes in the introduction, and it’s true: his fingerprints are on every page.

From those hundred, I picked out half a dozen to share here. Each is for two players; each requires only pens and paper; and each has surprising strategic depths to plumb.

May the wondrous light of Joris Radiation shine upon you in these strange times!

(*See the bottom of the post for an interview with Walter.*)

**1.**

Magical Squares

Magical Squares

In an actual magic square, every row, column, and diagonal has the same sum. In Walter’s game, you won’t achieve that, but the goal is to get as close as possible.

Each player begins with a blank square, then secretly places numbers in the four corners. You may use whatever numbers you like (including repeats).

Then, you reveal your squares to each other. Now, whatever numbers your opponent put in her *corners*, you must put in your *edges (*in whatever order you like).

Finally, you can choose whatever number you like for the center.

The goal: have as many rows, columns, and diagonals as possible sharing the same sum.

A pair with the same sum scores 1 point; a trio scores 2 points; a quartet scores 3 points; and so on. (This sample game ended in a 3-3 tie.)

What happens if you start with identical numbers in your corners (e.g., 7, 7, 7, 7)? What if you pick radically different numbers (1, 10, 100, 1 million)? Is there a best strategy? If so, is it deterministic or probabilistic? The field is open for exploration!

**2.**

**Bunch of Grapes**

This is, in one sense, a standard game of territory control. It’s like dozens of others I’ve encountered. Yet I can’t get it out of my head. It’s something about those juicy grapes, the silly theme, and the lovely drawings that result.

First, draw a bunch of grapes. Make it clear which grapes share a border.

Then, by turns, each player picks a grape on which their “fly” begins, and marks it with a colored dot.

Then, take turns moving. (Whoever placed their fly second should begin.)

On each move, your fly consumes the grape it’s on (shown by fully coloring in the grape), then moves to an adjacent grape.

Whoever winds up unable to move, because there are no adjacent grapes available, is the loser. Here, three more moves have been made:

The strategy seems straightforward, but the grapes can trick the eye, lending an element of suspense. (You may have less territory left than you think!) Also, whereas most pencil-and-paper games leave the paper coated with crisscrossing gibberish, this one ends up like a page from a coloring book.

Here, purple wins! (Green fly made some bad life choices.)

Ideally played while snacking on grapes.

**3.**

**Black Hole**

This has more the flavor of a puzzle than a game. It’s a puzzle I’ve yet to solve.

You begin by drawing a pyramid of 21 circles. Draw six on the bottom row, five on top of that, four on top of that, and so on.

Then, take turns writing a 1 in a circle of your choice.

After that, take turns writing 2, 3, and so on, in order. (You must write your numbers in order; no skipping ahead.)

When you have each written your 10, there will be one circle left blank: the black hole.

The black hole destroys all its neighboring circles. Whoever has a greater sum of numbers left over – that is, whoever loses a smaller sum to the black hole – is the winner.

If this speedy game is still too slow for your overheated 21st-century attention span, then you can eliminate the bottom row, and play using just 15 circles, where each player writes the numbers from 1 to 7.

(Or if you want to kill an extra ten minutes, add three more rows, for a total of 45 squares, so that each player writes the numbers from 1 to 22.)

Anyway, after a few rounds, I still have very few strategic intuitions about this game, but I love the simplicity of the design.

**4.**

**Imparium**

You begin with a 6 by 6 grid. Players take turns filling in pairs of adjacent squares, as if covering them with a domino.

These covered squares belong to nobody. Rather, you are fighting for control of the *other* squares, which you claim **when your domino completes a fence closing off a region with odd number of squares**.

Close off 2, 4, 6, or 8 squares? That’s useless. Close off 1, 3, 5, 7, or 9? You can claim them. (More than that, though, and it doesn’t count.)

The winner is whoever claims more squares by the end.

Some games, like “Bunch of Grapes” and “Black Hole,” feel so simple that they must have existed all along, as if their designer merely “discovered” them. Other games, like this one, are so quirky that they can only have been invented.

Or perhaps I should say “bred.” To my eye, this game has genetic traces of Nim, Cram, Dots and Boxes, and more.

**5.**

**Snake Fight**

Many pencil-and-paper games feel airy and abstract. Why dots? Why boxes? Why tics, tacs, and toes? That’s why I love the cute and highly literal theme of this one; the game resides right there in the name.

Snake fight!

Begin with a 5-by-5 array of dots, and draw in the outline. The players begin their snakes in opposite corners.

The goal is to cross the enemy snake as many times as possible.

You take turns extending your snake via vertical, horizontal, or diagonal lines. Your snake can never cross or touch itself, and cannot trace over a segment that has already been drawn (by you, your opponent, or the border).

The game continues until neither player can move. Make sure to keep score as you go; otherwise, you won’t be able to tell who crossed whom!

(Fine print: Passing through the enemy’s “head” counts as a crossing. So does the last move above for orange, where the head reaches the enemy, but doesn’t pass through.)

**6.**

**Collector**

I’ve saved for last the game that is perhaps the simplest – not to mention the deepest.

Begin with a 6 by 6 grid. On each turn, you mark any box you like, but you must also eliminate an empty neighboring box.

Eliminating a diagonal neighbor is allowed.

The winner is whoever creates the largest group of connected marks. (Diagonal connections count.)

Play until no more moves are possible.

I have no idea why this one is called “Collector.” Why not “Connector”? Or “Barrier”? No matter: a pleasant mystification is a natural byproduct of Joris Radiation.

The gameplay here reminds me of Amazons, a classic 1988 territorial game in which each move involves the annihilation of a square. This creates a “the world is falling away!” flavor of drama.

### A Brief Interview with Walter

**How did you develop an interest in designing games?**

I always liked board games. And in my youth, there was a kind of a culture in pen paper games, long before computers were there. And then there was Martin Gardner and his math puzzles, which fascinated me.

**How do you go about designing a game? What’s the process like?**

The first ones are the hardest. But once you give your mind the task: “invent games”, it obeys. It starts to be creative. And then, whenever there is a kind of inspiration, a special pattern you see, a combination, your creativity will turn it into a game.

**Where do you farm for ideas? As in: what board games do you play? What books do you read?**

Well, mathematical puzzles of course, but also puzzles and pastimes for children. Also: existing board games in the world, and there is a lot of them. I myself play Go. And reading, I’ve read an enormous amount of books. Must be thousands. All kind of genres, philosophy, science fiction, novels, also the classical ones, from Dostoevsky to Rimbaud. Strange books likes the ones of Madame Blavatsky. Dali. Marinetti, Dada…Popular science also, but mostly in magazines. Art, history, cultural history…. I speak Dutch, French, English, German, and a bit of Spanish, In the first 4 languages I can read books rather easily. So, living in Belgium, in the middle of those cultures, you can imagine what an enormous amount of books and rare books you can find. With the fast train, you are from Brussels in about 1 hour in Paris, 1,5 in London, Cologne in Germany. Where I live, I can go by tramway from the Netherlands to France.

**Do you think of your games as fundamentally mathematical?**

Yes, I think of all games as fundamentally mathematical. I call myself a “matheist.”

**Lots of games in your book are adapted from board games. How do you decide if a game is suitable for adaptation?**

It has to be able to become a pen and paper game. And that is fundamentally different. In material board games, it’s all about the empty spaces; in a pen and paper game, about the occupied ones.

**What are your favorite games (not of your own making)?**

I played them all once. Now, I’ve limited myself to only Go.

**What are your favorite games (of your own making)?**

Since I invented them, I like them all. But Sequentium is definitely my favorite.

Thanks to Walter for answering my questions. You can find his work on Facebook, at Deviant Art, and of course in his book.

And at his request, here’s Walter’s explanation of his crown jewel, Sequencium:

Do you know of any website or the like where these can be played online? I’d love to play these with my friends but lockdown prevents me from meeting them.

Unfortunately, no – as far as I know these games haven’t been posted online before.

Maybe someone enterprising who’s reading this comment wants to put together electronic versions?

Magic Squares could probably be played via Skype or Zoom; the others you might manage with screen-sharing, but it’d be clunky at best.

I made a website with Sequencium (local and online multiplayer). There are some bugs but it works well enough to have fun.

Here’s the website: https://sequencium-server.herokuapp.com/

(May be slow to load at first because the server is not always awake.)

And here’s an example video of multiplayer (as well as the code): https://github.com/SirFurness/Sequencium

This is wonderful – thanks for making it!

Regarding “draw a chair with one continous line”, Picasso did many one line drawings, which are worth perusing.

Ben, these are wonderful. Thank you so much for flagging Walter Joris’s work! I love these types of games and am shocked that I hadn’t known about him before.

If there is something I (and my family of child play testers) could do to help with your book, feel free to contact us (jgr & first 3 digits of pi & usual gmail stuff is my email). We are enthusiastic matheists as well.

We will look out for Joris in our online Go matches, too (we play on kgs and OGS mostly).

Hello! I’d REALLY like to purchase a copy of this book!! But the cheapest once is $923!! Do you know how I can order a copy (at normal price? )

Your article is SO, SO cool… THANK YOU!+

Yeah, I’m afraid this post shined a bit of a spotlight on the book, which meant that the few copies available for sale skyrocketed in price!

I don’t know of a good way to get a normal-priced copy, although you can check out Walter’s social media accounts (linked at the end of the post) where he’s posted a number of other games. I’ll also try to get around to posting a second installment of Joris games sometime in the next few months. (Not as good as having the book itself, I realize!)

I’m pretty sure that Sequencium, as is, is a draw (I think I can weakly solve it)

Consider Sequencium, except both players have the ability to “pass”(i. e. lose their turn).

Assume the second player has a winning strategy. Then the first player passes and uses the second player’s winning strategy, contradiction.

The first player doesn’t have a winning strategy either, as the second player can copy all of the first player’s moves with a 180 degree rotation.

Finally, passing is never optimal unless forced, in the sense that there is always a non-passing move at least as good as passing.

Therefore Sequencium is a draw, strategy being the second player’s 180 degree rotation symmetry.

Yeah, I think this is true! For this reason my preferred way to play Sequencium is with a “1-2-2-2” move pattern, so that the first player moves once, and after that, every move is a double-move.