CHOMP

Main Line Math Circle

January 21, 2026

Bill Hawkins

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Welcome!

• As you enter be sure to check in!
• Sit down anywhere you’d like - but please don’t move the chairs to form large groups.
• Introduce yourself to group mates as they arrive.
• We’ll get started soon - in the meantime give the problems a try!

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Thanks!!!

• Support for the Main Line Math Circle is provided by the Mathematical Association of America and the Mary P. Dolciani Halloran Foundation.
• Thanks to the Villanova Department of Mathematics and Statistics for the space.
• Thank you to our volunteer teachers: Mr. Hawkins (director), Mrs. Hawkins, Mr. Vaccaro
• Thank you to our high school mentors!
• Thanks to our Villanova volunteers: Dr. Haymaker (director), VU students
• Thank you all for coming out.

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Schedule

• 5:00 - 5:20 Warm Up Problem
• 5:20 - 5:50 Pizza!
• 5:50 - 6:10 Play Chomp (2xN game)
• 6:10 - 6:30 Strategy Discussion
• 6:30 - 6:50 Play Chomp (NxN game)
• 6:50 - 7:00 Strategy Discussion
• 7:00 - 7:15 General Case (MxN game)

Warm Up Problem

SOLUTION

The answer is 83. Here is a solution that involves working backward from the final result of 1.

For each stage, we have a collection of integers and we set these integers equal to to find the integer(s) at the previous stage.

SOLUTION

We can both check and visualize the solution using a tree diagram:

2xN Chomp

Chomp is a strategic two-player game played on a rectangular grid resembling a chocolate bar.

2xN Chomp

Players take turns selecting a square to “chomp” (eat).

When you chomp a square you remove that square and all squares positioned to the right and below it.

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2xN Chomp

Here’s the catch…the top left square is poisoned. The player compelled to eat this square loses.

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2xN Chomp

Suppose Player 1 chomps the square in the (1,10) position.

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Which squares are eliminated?

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2xN Chomp

All squares to the right and below are removed.

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Now, suppose player 2 chomps (2,5). What happens?

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2xN Chomp

There is nothing below (2, 5) so all the squares to the right are removed.

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Suppose Player 1 chomps (1,2)…good idea?

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2xN Chomp

Bad idea…we wish to force the other player to eat the poison. By selecting (1,2), player 2 can eat square (2,1) forcing player 1 to eat the poison and lose.

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2xN Chomp

Does Player 1 or Player 2 have an advantage?

Describe a winning strategy for that player

Work Backwards

Work Backwards

?????????

A Strategy!

NxN Chomp

Does Player 1 or Player 2 have an advantage now?

Can you describe a general strategy for that player that will win every time?

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NxN Chomp

Player 1 should NOT take only the square in the bottom right corner!

While this creates symmetry, it leaves open a counter-move for Player 2 that does the same.

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NxN Chomp

Player 1 should take the square in the (2,2) position!

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NxN Chomp

Player 1 should take the square in the (2,2) position!

This creates a situation where Player 2 must pick from a row or column and Player 1 can mirror their move.

MxN Chomp

Playing CHOMP on different size boards can be fun because the exact winning strategy for these boards isn’t easily known or easily discoverable.

However…we CAN prove the existence of such a strategy!!!!

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SURVEY!!

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• Further reading:
• The Collatz Conjecture (The Unsolved Problem behind tonight’s warm-up)
• Veritasium Video
• Try different starting points
• CHOMP
• Play online
• Read about the history of the game here
• For more math circle: Join us on February 25th!