## Arithmetical Tennis

Louigi Verona

November 2020

I have always been fascinated by games that can be played in the mind alone: no equipment required. An image of players engaged in a battle of wits on an invisible playing board kept fueling my inspiration.

Strictly speaking, any perfect information game can in theory be mastered to be played this way. Experienced chess players can run through full games without a physical board. Very straightforward techniques exist to teach that ability to virtually anyone.

However, the problem with blindfold chess is that it is not readily accessible. Unless someone has committed a significant amount of effort to mastering the visualization of the board and pieces, practiced the memorization of 10, 20, 30 moves, and has an opponent who has been as committed, such a game would not be possible.

My intention was to come up with a game that would be accessible to anyone and would require no special preparation, while being non-trivial and offer a mental challenge. Several years ago I came up with a promising mechanic, but it actually took some time to turn it into a playable game.

Thus, "Arithmetical Tennis" was born. Online version can be found here.

**Gameplay and rules**

The game involves saying numbers at each other and would probably sound complicated and cryptic to an outside observer. Here's a typical "Arithmetical Tennis" match:

- Tom's serve: 76
- Bob: 4
- Tom: 1562
- Bob: 81
- Tom: 9
*Point for Bob!* - Tom's serve: 1
- Bob: 1
- Tom: 2
- Bob: 35
*Point for Tom!* - Bob's serve: 84
- Tom: 3718
*Point for Bob!* - Tom's serve: 9887
- Bob: 64
- Tom: 1562
- Bob: 81
- Tom: 9
*Point for Bob!* - Game!
- Bob wins 3:1

However, the game is governed by very simple rules and should not be difficult for anyone who can do basic addition.

Arithmetical tennis is played between two players and involves producing a sequence of digits based on a simple principle: the next digit is the sum of the previous two. However, unlike the Fibonacci sequence, if you go over 9, you must do a digit sum of the result in order to still end up with a single digit. So, 13 becomes 1+3=4, for example.

There is also a rule that governs the amount of digits one must respond with. If the last digit of your opponent's move ends with 1, 2, 3 or 4 - you must respond with a number consisting of 1, 2, 3 or 4 digits. If it ends with 5, 6, 7, 8 or 9, you can respond with any amount of digits, but never more than 4.

The match starts with a first serve, which should always be a two-digit number. The opponent should then respond with a legal move, based on the rules outlined above. If the response ends in 5, 6, 7, 8 or 9, then the return was unsuccessful and the serving player receives a point. If, however, the response ends with 1, 2, 3 or 4 - in other words, a move that forces a certain amount of digits on your opponent - the return was successful and the play continues.

If a point is lost, the player who lost the point serves. Note that subsequent serves can be a four digit number or lower, only the first serve is limited to a two-digit number.

If a player gives a response that breaks any of the rules, it is a "fault". This moves the serve to their opponent and also gives her 2 points. The incorrect move is not considered and the serve must continue from the last legal move.

The game ends after the sequence loops around. If this happens in the middle of a serve being played, the serve should be completed.

The sequence will always loop. In the example above, the sequence is: 764156281911235843718988. Any pair of numbers in that sequence could be taken as a starting point and be used to produce the whole loop.

Now that we know the rules, let's go through our match example.

- Tom's serve: 76 Tom posts a 2-digit number
- Bob: 4 Since 76 doesn't end with 1, 2, 3 or 4, Bob can choose to respond with any 4-digit number or lower. His options include 4 or 41. Choosing 415 or 4156 would lose a point, since those numbers end in digits larger than 4.
- Tom: 1562 Tom is forced to respond with a 4-digit number, due to Bob posting 4. But his number luckily ends with a 2, throwing the ball back on Bob's side of the court.
- Bob: 81 Bob is forced to respond with a 2-digit number
- Tom: 9
*Point for Bob!*Tom is forced to respond with a single digit number and it turns out to be a 9, which is tantamount to not returning the ball and losing a point - Tom's serve: 1 Since Tom lost a point, he is given a chance to serve. Since this is not a first serve, he is no longer limited to a 2-digit number.
- Bob: 1
- Tom: 2
- Bob: 35
*Point for Tom!*What an exchange! And a beautiful win for Tom! - Bob's serve: 84
- Tom: 3718
*Point for Bob!* - Tom's serve: 9887 Tom makes a mistake here. He could have played 9, leaving Bob no options, since the next four digits in the sequence lose him a point
- Bob: 64 The players loop the sequence: it started with 76. Since we are in the middle of an ongoing play, we have to finish it.
- Tom: 1562
- Bob: 81
- Tom: 9
*Point for Bob!* - Game!
- Bob wins 3:1

**1. Analysis**

**1.1 Mental operations.**

Let's first analyze the game from it's applicability to being played with no equipment.

In order to make a move, players need to only remember the previous two numbers. This makes it easy to make your move and to follow the response of your opponent. Doing it quickly and without mistakes might, perhaps, require a bit of practice, but no practice is required to start playing. One can even run a match with oneself and discover that it is simpler than it seems.

There are some overarching things to remember, specifically the overall score and the initial serve, but the score rarely goes over 2. The length of matches, apart from a single sequence 39336966, is always the same and can be felt intuitively after just several plays.

**1.2 Depth.**

For inexperienced players, the game might provide a lot of entertainment value: starting with a random 2-digit number and seeing where it leads.

But the game also gives players ample space for improvement, both in terms of execution and tactics. An experienced player would not only provide answers correctly and quickly, but also learn to enhance one's mental vision, calculating more digits ahead and making better decisions. An "Arithmetical Tennis" master might learn optimal plays for many sequences.

Here's an example of how seeing more digits ahead can be an immediate advantage:

- Tom: 15
- Bob: 62
- Tom: 81
- Bob: 9

- Tom: 15
- Bob: 6281
- Tom: 9

It's easier to come up with 62 as a response, but responding with 6281 is the winning move. It requires to not only come up with the four digits, but also calculate the fifth digit and see that it's going to be a 9.

**1.3 Limitations.**

A game designed to be played with no equipment is bound to be limited in its theoretical complexity and depth, simply due to the limits of human operational memory. "Arithmetical Tennis" shines as a mental game, but is less interesting, although not trivial, when played on paper or on a computer.

There are 5 unique sequences that cover all possible pair combinations, of which one is not playable and the other is very short:

- 191123584371898876415628
- 292246178652797753821347
- 39336966
- 494483257314595516742685
- 99

But the situation is not too bad: playing the same sequence from different entry points would yield very different results. So, "Arithmetical Tennis" does offer a considerable supply of variability.

On the downside, sequences will have deterministic bits. It's therefore not always possible to turn one's fortunes around in case of perfect play. In our 76 sequence, there is no way for Tom to win. He can only lose less. Worst scenario is 1:4, best is 2:3, as follows:

76

41

5 0:1

6281

9 1:1

112

35 1:2

843

718 2:2

9

8876 2:3

So, the game is far from fair and who goes first could be very important.

This is not true for all pairs, though, and many sequences end up being really close or draws. Try sequence 51, for instance.

Fairness can be improved by coming up with sets of sequences which are draws, so that you lose only if you make a mistake. Or selecting sequences where the starting player is poised to win, but taking turns serving first.

But again, the game was designed for mental play, so I am not particularly worried: the game has enough depth to be engaging for human players, ample room to improve, and getting to perfect play is actually pretty difficult.

**Miscellaneous.**

Although it's easier to give responses digit by digit, if you consider yourself to be a professional arithmetical tennis player, you should instead respond with proper numbers. Instead of saying "two, three, five", say "two hundred thirty five". This could also be lots of fun if you want to impress people who don't know the game!

**mtanzer**, whom my readers might know as the best Perfectionist solver, noticed that if you arrange a sequence into a circle, numbers across will add up to 9 (except for 9). Take, say, sequence A from that list above, if you write it down in two rows, while the 9s are in front of each other, the rest of the numbers all add up to 9: 8+1, 7+2, 6+3, etc.:

191123584371

898876415628

0 is not used in the game. If you do choose to use it, you absolutely can, but you'd quickly discover that its behavior makes it identical to a 9.

When researching similar games before writing this article, I stumbled upon a game devised by Lewis Carrol in 1872 called "Arithmetical Croquet". It features strikingly similar game design decisions of not requiring too many things to keep in mind, as Carrol also devised it to be played with no equipment. I invented my game independently, but decided to call it "Arithmetical Tennis" in homage to "Arithmetical Croquet".