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Reference

The Arcade

Every game and tool on the site, in chapter order. Each link jumps straight to the board. Robo is waiting.

Chapter 1 · What is a game?
Rock, Paper, Scissors
Robo picks at random.
Chapter 1 · What is a game?
Rock, Paper, Scissors, Lizard, Spock
Rows are you, columns are Robo. Hover a row to count its wins.
Chapter 1 · What is a game?
Build a fair game
Green = row beats column. Red = row loses. Click to flip.
Chapter 2 · The Cookie Game
Share or Grab?
You pick the row. Robo picks the column.
Chapter 2 · The Cookie Game
Find the best responses
Down a column for red. Across a row for blue.
Chapter 2 · The Cookie Game
Cross it out
Rows are your choices. Columns are Robo's.
Chapter 3 · Nobody wants to switch
Equilibrium hunt
Five games, five different answers.
Chapter 3 · Nobody wants to switch
Best-response arrows
Red arrows: where you would move. Blue arrows: where Robo would move.
Chapter 3 · Nobody wants to switch
Equilibrium counter
Red numbers are yours, blue are Robo's. Ties count.
Chapter 4 · Being unpredictable
Matching Pennies
+1 to you if different, +1 to Robo if the same.
Chapter 4 · Being unpredictable
The odds of the odds
Probability is area: width times height.
Chapter 4 · Being unpredictable
The penalty kick
Numbers are the chance of a goal, in percent.
Chapter 5 · Thinking backwards
The 21 Game
Take 1, 2, or 3 on your turn. Last stone wins.
Chapter 5 · Thinking backwards
Label the positions
W = the player whose turn it is can force a win. L = they cannot.
Chapter 6 · Nim and binary
Nim
Take any number of stones from one pile. Last stone wins.
Chapter 6 · Nim and binary
Base 2
Type a number, or click the boxes to switch places on and off.
Chapter 6 · Nim and binary
Nim-sum by hand
Click each answer box: 1 if the column has an odd number of 1s, otherwise 0.
Chapter 6 · Nim and binary
Find the winning move
Click a stone to choose your move, then check it.
Chapter 7 · Seeing the whole tree
Tic-tac-toe
You are X. Robo is O.
Chapter 7 · Seeing the whole tree
Tree puzzle
X to move. Which square?
Chapter 7 · Seeing the whole tree
Three kinds of first move
Symmetry shrinks the tree.
Chapter 8 · A proof that you can win
Chomp
Do not eat the poison.
Chapter 8 · A proof that you can win
How many shapes?
A bar with r rows and c columns has C(r + c, r) shapes.
Chapter 9 · Hex
Hex
6 by 6. Red joins top to bottom. Blue joins left to right. No draws.
Chapter 9 · Hex
Fill the board
Every full board has exactly one winner. Find the chain, or try to build a board without one.
Chapter 9 · Hex
Inside Robo's head
The five best cells from Robo's latest move in the game above, with the counts that chose them.
Chapter 10 · Every game is secretly Nim
The mex calculator
Type a set. The smallest missing whole number lights up.
Chapter 10 · Every game is secretly Nim
Label the positions with Grundy values
G(n) = mex of the values one move away. G(0) = 0.
Chapter 10 · Every game is secretly Nim
Two games at once
Move in one game per turn. If you cannot move anywhere, you lose.
Chapter 10 · Every game is secretly Nim
Kayles
Knock down one pin, or two standing next to each other. Last move wins.
Chapter 11 · Playing again and again
Twenty rounds of the Cookie Game
Share or Grab each round. Both share: 3 and 3. A grab on a share: 5 and 0. Both grab: 1 and 1.
Chapter 11 · Playing again and again
Round robin
Every strategy plays every other one, and itself. Highest total wins.
Chapter 12 · I cut, you choose
Cut and choose
You cut. Robo picks the piece it likes better. You get the other one.
Chapter 12 · I cut, you choose
The moving knife
Shout Stop when the piece left of the knife is worth one third to you. First to shout takes it.
Chapter 12 · I cut, you choose
Envy detector
Rows: how each player values each piece, adding up to 100. Everyone got the piece with the yellow frame.
Chapter 13 · Going once, going twice
The English auction
Price climbs by $5. Last bidder standing pays the last price called.
Chapter 13 · Going once, going twice
First price, ten rounds
Highest bid wins and pays its own bid. Robots bid 3/4 of their value.
Chapter 13 · Going once, going twice
Why honesty wins
Second price: if B beats R you win and pay R. Ties go to the other bidder.
Chapter 13 · Going once, going twice
Second price, ten rounds
Highest bid wins and pays the second-highest bid. Robots bid their true values.
Chapter 13 · Going once, going twice
The falling clock
Price falls $1 at a time. First to shout takes it at that price.
Chapter 14 · How to pick a winner when nobody agrees
The Fish group's trick
Rule: plurality. Most first-place votes wins.
Chapter 14 · How to pick a winner when nobody agrees
Find the trick
Rule: Borda count. 2 points for a first place, 1 for a second, 0 for a third.
Chapter 15 · Take it or leave it
The ultimatum: you propose
10 coins. You offer a split. Robo says yes or no. No means nobody gets anything.
Chapter 15 · Take it or leave it
The ultimatum: Robo proposes
Robo offers you some coins. Yes: the split happens. No: nobody gets anything.
Chapter 15 · Take it or leave it
Five pirates, 100 coins
At least half the votes, counting the proposer's own, and the split happens.
Chapter 15 · Take it or leave it
The melting cake
Worth 100 now. Each round without a deal, it loses 20 percent. After the last round, nothing is left.
Chapter 16 · Hawks and doves
Two ways to fight over a worm
Both animals are the same kind of animal, so the grid is symmetric.
Chapter 16 · Hawks and doves
Payoff versus p
Hawks spread where the red line is higher. Doves spread where the blue line is higher.
Chapter 16 · Hawks and doves
Let it evolve
300 animals. Random pairs. Babies in proportion to points.
Chapter 16 · Hawks and doves
Three lizards
Orange beats blue, blue beats yellow, yellow beats orange.
Chapter 18 · What everybody knows that everybody knows
The muddy children
You are one of the children. Robots reason perfectly and never peek.
Chapter 18 · What everybody knows that everybody knows
The knowledge ladder
Every robot shown is muddy. Which rungs hold?
Chapter 18 · What everybody knows that everybody knows
Two thirds of the average
You and nine Robos. Closest to two thirds of the average wins.
Chapter 18 · What everybody knows that everybody knows
Red hat or blue hat?
You are a player. The two Robos follow the clever plan. Hat colors here are just hat colors.
Chapter 20 · The pond that everyone owns
The pond
Four fishers, 20 seasons. The pond regrows after every catch.
Chapter 20 · The pond that everyone owns
What if everyone did that?
All four fishers take the same amount, every season, for 20 seasons.
Chapter 20 · The pond that everyone owns
The coin pot
10 coins each. The pot is multiplied by 1.6 and split four ways.

Rock, Paper, Scissors

Rock, Paper, Scissors, Lizard, Spock

Build a fair game

Share or Grab?

Find the best responses

Cross it out

Equilibrium hunt

Best-response arrows

Equilibrium counter

Matching Pennies

The odds of the odds

The penalty kick

The 21 Game

Label the positions

Nim

Base 2

Nim-sum by hand

Find the winning move

Tic-tac-toe

Tree puzzle

Three kinds of first move

Chomp

How many shapes?

Hex

Fill the board

Inside Robo's head

The mex calculator

Label the positions with Grundy values

Two games at once

Kayles

Twenty rounds of the Cookie Game

Round robin

Cut and choose

The moving knife

Envy detector

The English auction

First price, ten rounds

Why honesty wins

Second price, ten rounds

The falling clock

The Fish group's trick

Find the trick

The ultimatum: you propose

The ultimatum: Robo proposes

Five pirates, 100 coins

The melting cake

Two ways to fight over a worm

Payoff versus p

Let it evolve

Three lizards

The muddy children

The knowledge ladder

Two thirds of the average

Red hat or blue hat?

The pond

What if everyone did that?

The coin pot