# Chapter 4: The Elder (puzzle hints and solutions)

#### Puzzle 1 – Twenty-one

You begin at 0 and each player may count up 1, 2 or 3 numbers on their turn. Then the next player does the same and so on. Whoever is forced to say 21 loses the game.

If Terry goes first and says 1, can you think of a strategy to ensure you win? Think about how Terry knew he’d won when he reached 16.

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#### Puzzle 2 – Nuts

“The aim of the game is simple. Each player takes their turn to place a nut on this leaf. You may not place a nut on top of another and it must lie wholly within the leaf. The last player to be able to place a nut is the winner. As the challenger, I will let you place the first nut.”

Can you find a way of ensuring your victory, no matter how Chris plays?

#### Hint 1

“Think back to our games of 21. What was the winning strategy? How were you sure you could always reach a multiple of 4 and is there a similar property to consider in this game?”

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#### Hint 2

“In 21, you knew no matter what I said, you could count up enough to reach a multiple of 4. In some sense, you did the opposite to my action every round, adding 4 minus whatever I added. Can you do something similar here?”

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#### Hint 3

“There is something special about the playing board and nuts. Can you exploit it? Is there a simple strategy you can follow which tells you where to place your nut once Chris has placed his? Where must you place the first nut to ensure this always works?”

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#### Puzzle 1 (twenty-one) – Solution

In the game of 21, the key is to always reach a multiple of 4 at the end of your go. Since each player must add either 1, 2 or 3 to the total, the second player can always add 4 – [player one’s addition] to reach 4, then 8, 12, 16 and ultimately 20. The second player has the winning strategy.

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#### Puzzle 2 (nuts) – Solution

Each game may take a hugely different path and the final nut could be placed on any of infinitely many possible board states. Luckily, there is a simple strategy we can construct which will work in any game.

The ‘opposite’ which Terry means to consider is the symmetry in the game. We need a way to conserve this symmetry at the end of every two goes. Thus, we will place a nut directly opposite of wherever Chris places his. An example is shown below:

The only time this can fail, is if where we must place a nut overlaps with where Chris has placed his. This can only happen near the centre of the board, and thus to avoid this, we place the first nut in the exact centre. Now whenever Chris can place a nut, so can we. Thus we have our winning stratagem.

This game and strategy works on any board which has rotational symmetry of order 2, such as a rectangle. The strategy is always to place your nut as if rotating the opponent’s move by 180° about the board’s centre.

There is an interesting theorem involving games between two players similar to 21. In a finite, perfect information game, where each player knows everything there is to know about the game at any time, there always exists a non losing strategy. Read more about this here.

Matt Parker has also done a great video about a game which works much like 21, which you can watch here.