How  to always win (at some games)

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Race to 100

1. Split into pairs

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Race to 100

1. Split into pairs

2. Decide who goes first

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Race to 100

3. The first person starts by saying a number from 1 to 10

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Race to 100

3. The first person starts by saying a number from 1 to 10

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4. The other person adds to it any number from 1 to 10

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Race to 100

5. Go back and forth adding any number from 1-10

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Race to 100

5. Go back and forth adding any number from 1-10

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Race to 100

5. Go back and forth adding any number from 1-10

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Race to 100

5. Go back and forth adding any number from 1-10

18

28

35

44

51

...

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Race to 100

5. Go back and forth adding any number from 1-10

18

28

35

44

51

...

6. The person who reaches 100 first WINS!

100

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Race to 100

5. Go back and forth adding any number from 1-10

18

28

35

44

51

...

6. The person who reaches 100 first WINS!

100

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Race to 100

Does anyone have any ideas how you can improve your chances of winning?

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Race to 100

Let us work backwards....

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Race to 100

Let us work backwards....

If your opponent says a number bigger than or equal to 90, you automatically win. Why?

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Race to 100

Let us work backwards....

If your opponent says a number bigger than or equal to 90, you automatically win. Why?

100

90

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Race to 100

Let us work backwards....

If your opponent says a number bigger than or equal to 90, you automatically win. Why?

100

90

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Race to 100

So how can you force your opponent to say a number bigger than or equal to 90?

100

90

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Race to 100

So how can you force your opponent to say a number bigger than or equal to 90?

100

89

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Race to 100

So how can you force your opponent to say a number bigger than or equal to 90?

100

89

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Race to 100

If you stop at 89, your opponent can reach 99 at max, and then you can win!

100

89

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Race to 100

If you stop at 89, your opponent can reach 99 at max, and then you can win!

100

89

89

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Race to 100

Let us use the same logic again.

100

89

89

79

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Race to 100

Let us use the same logic again.

100

89

89

79

78

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Repeating this logic tells us that the race is made out of shorter races

100 to 89 to 78 to 67 to 56 to 45 to 34 to 23 to 12 to....

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Repeating this logic tells us that the race is made out of shorter races

100 to 89 to 78 to 67 to 56 to 45 to 34 to 23 to 12 to....

1! So we begin with a race to 1. What does this mean?

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Repeating this logic tells us that the race is made out of shorter races

100 to 89 to 78 to 67 to 56 to 45 to 34 to 23 to 12 to....

1! So we begin with a race to 1. What does this mean?

It matters who starts. If you start, then you can win by using the strategy.

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Race to 100 strategy

1. Be the one to start and start by saying 1.

2. Always make the numbers

100, 89, 78, 67, 56, 45, 34, 23, 12

3. Win! :D

The game of Nim

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The game of Nim

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We start with two equal piles of beads

The game of Nim

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Decide who goes first. Take turns playing.

The game of Nim

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In each turn, choose ONE pile and take as many beads as you want.

The game of Nim

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In each turn, choose a pile and take as many beads as you want.

The game of Nim

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In each turn, choose a pile and take as many beads as you want.

The game of Nim

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In each turn, choose a pile and take as many beads as you want.

The game of Nim

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In each turn, choose a pile and take as many beads as you want.

The game of Nim

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In each turn, choose a pile and take as many beads as you want.

The game of Nim

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In each turn, choose a pile and take as many beads as you want.

The game of Nim

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In each turn, choose a pile and take as many beads as you want.

The game of Nim

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Person who takes the last pile wins. Or, if you have nothing to pick, you lose!

The game of Nim

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Let's play!

Rules:

1. Decide who goes first. Take turns.
2. Pick a pile, take any number of beads
3. Person who take the last beads wins

The game of Nim (strategy)

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What ideas do you have?

The game of Nim (strategy)

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We will again work backwards.

The game of Nim (strategy)

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We will again work backwards.

What happens when you finish the first pile?

The game of Nim (strategy)

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We will again work backwards.

What happens when you finish the first pile?

The game of Nim (strategy)

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We will again work backwards.

What happens when you finish the first pile?

The game of Nim (strategy)

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We will again work backwards.

What happens when you finish the first pile?

You lose! Your opponent can pick the other pile.

The game of Nim (strategy)

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We will again work backwards.

What happens when you finish the first pile?

You lose! Your opponent can pick the other pile.

The game of Nim (strategy)

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So we should force our opponent to pick the first pile.

The game of Nim (strategy)

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So we should force our opponent to pick the first pile.

This can only happen when the piles are (on your opponent's turn)

The game of Nim (strategy)

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And how can we make 1-1 happen? Our opponent must have no other choice

The game of Nim (strategy)

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And how can we make 1-1 happen? Our opponent must have no other choice

The game of Nim (strategy)

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So we win if on our opponent's turn the piles are equal

The game of Nim (strategy)

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So we win if on our opponent's turn the piles are equal

This is called the copying strategy.

1. Take the second turn.

2. "Do whatever your opponent does".

The unequal game of Nim

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We start with two piles but not both equal.

The unequal game of Nim

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What is the strategy now?

The unequal game of Nim

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Using the idea from the equal version: "one who makes the piles equal wins"

The unequal game of Nim

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Using the idea from the equal version: "one who makes the piles equal wins"

The unequal game of Nim� (strategy)

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1. Take the first turn.
2. Make the piles equal
3. Use the copying strategy now

​

In math, this is called reducing one game to the other.

Another variant of Nim�(Wythoff's game)

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Start with unequal piles

Another variant of Nim�(Wythoff's game)

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Same rules as Nim with an extra rule:

You can take an equal number of beads from both piles at once.

​

Another variant of Nim�(Wythoff's game)

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Same rules as Nim with an extra rule:

You can take an equal number of beads from both piles at once.

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Another variant of Nim�(Wythoff's game)

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Let's play!

Rules

1. Decide who goes first. Take turns.​
2. Pick a pile, take any number of beads​. Or take an equal number of beads from both piles.
3. Person who take the last beads wins​

​

Another variant of Nim�(Wythoff's game)

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Why doesn't our previous strategy work?

Another variant of Nim�(Wythoff's game)

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Why doesn't our previous strategy work?

If you make the piles equal, you lose!

Another variant of Nim�(Wythoff's game)

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Why doesn't our previous strategy work?

If you make the piles equal, you lose!

A new game?

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Before we discuss the strategy, let's talk about another game called the Queen's game.

A new game?

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You start somewhere on the right side of the chessboard

A new game?

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You have to reach the other corner

A new game?

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You can move like a queen

Left

Down

Diagonally

as shown

A new game?

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Take turns playing

A new game?

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Take turns playing

A new game?

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Take turns playing

A new game?

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Take turns playing

A new game?

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Take turns playing

A new game?

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Take turns playing

A new game?

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Take turns playing

A new game?

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Take turns playing

Player 2 WINS!!

A new game?

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Let's play!

Rules:

1. Start on the right side and reach the bottom left corner.
2. You can only go left, down, diagonally left-down any number of squares.
3. One who reaches the corner first wins.

Before we discuss strategy...

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Did anyone notice something?

Both games are the same game!

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Let us take a look at how they are the same game.

Both games are the same game!

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Let us take a look at how they are the same game.

Remove from the right pile

Remove from the right pile

Remove from both

Both games are the same game!

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Both games are the same game!

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Both games are the same game!

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Both games are the same game!

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Both games are the same game!

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Both games are the same game!

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Now let us discuss the strategy

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90

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What did we learn today?

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1. Race to 100. Work backwards.
2. Nim and the copying strategy
3. Two games can be secretly the same, so they have the same strategy.

Did y'all have fun?

What did we learn today?

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1. Race to 100. Work backwards.
2. Nim and the copying strategy
3. Two games can be secretly the same, so they have the same strategy.

Winning is important but it's more important to have fun!