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The Chicken McNugget Problem

Main Line Math Circle

May 28, 2026

Bill Hawkins

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Welcome!

  • As you enter be sure to check in!
  • Sit down anywhere you’d like.
  • Introduce yourself to group mates as they arrive.
  • We’ll get started soon - in the meantime give the warm up problem a try!

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Schedule

  • 5:00 - 5:10 Introduce the McNugget Problem
  • 5:10 - 5:40 Frobenius Coin Problems - find the rule!
  • 5:40 - 6:10 Pizza & ?????
  • 6:00 - 6:10 Solve the McNugget Problem
  • 6:10 - 6:40 Variations
  • 6:40 - 7:00 Time to say thank you!

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Warm Up Problem

Chicken McNuggets come in boxes of 9 and 20. Suppose you were really hungry and wanted exactly 314 mcnuggets (and not a mcnugget more). How many boxes of 9 and how many boxes of 20 would you need to order?

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The Chicken McNugget Problem

Chicken Mcnuggets come in boxes of 9 and 20. If you wanted to order 21 nuggets, you’d be out of luck…Let’s call 21 an unattainable McNugget order.

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What is the largest unattainable McNugget order???

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Coin Problems

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Coin Problems

Country / Currency

Common Coin Denominations

United States (US Dollar)

1¢, 5¢, 10¢, 25¢, 50¢, $1

Eurozone (Euro)

1¢, 2¢, 5¢, 10¢, 20¢, 50¢, €1, €2

United Kingdom (Pound Sterling)

1p, 2p, 5p, 10p, 20p, 50p, £1, £2

Canada (Canadian Dollar)

5¢, 10¢, 25¢, $1, $2

Australia (Australian Dollar)

5¢, 10¢, 20¢, 50¢, $1, $2

Japan (Yen)

¥1, ¥5, ¥10, ¥50, ¥100, ¥500

China (Yuan / Renminbi)

¥0.1, ¥0.5, ¥1

India (Rupee)

₹1, ₹2, ₹5, ₹10

Brazil (Real)

5¢, 10¢, 25¢, 50¢, R$1

Mexico (Peso)

5¢, 10¢, 20¢, 50¢, $1, $2, $5, $10

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Group

Denominations

Largest Unattainable Amount

1

3¢, 8¢

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2

5¢, 8¢,

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3

4¢, 7¢

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4

5¢, 6¢

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5

3¢, 10¢

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6

4¢, 9¢

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7

3¢, 7¢

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8

5¢, 9¢

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9

6¢, 7¢

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10

6¢, 11¢

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A Formula?

Given two relatively prime denominations a and b, can you find a formula for the Largest Unattainable Amount?

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FROBENIUS

https://www.geogebra.org/m/wbhkhwx7

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Group

Denominations

LAU

How Many AUs?

1

3¢, 8¢

13¢

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2

5¢, 8¢,

27¢

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3

4¢, 7¢

17¢

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4

5¢, 6¢

19¢

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5

3¢, 10¢

17¢

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6

4¢, 9¢

23¢

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7

3¢, 7¢

11¢

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8

5¢, 9¢

31¢

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9

6¢, 7¢

29¢

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10

6¢, 11¢

29¢

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Group

Denominations

LAU

How Many AUs?

1

3¢, 8¢

13¢

7

2

5¢, 8¢,

27¢

14

3

4¢, 7¢

17¢

9

4

5¢, 6¢

19¢

10

5

3¢, 10¢

17¢

9

6

4¢, 9¢

23¢

12

7

3¢, 7¢

11¢

6

8

5¢, 9¢

31¢

16

9

6¢, 7¢

29¢

15

10

6¢, 11¢

29¢

25

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Another Formula?

Given two relatively prime denominations a and b, can you find a formula for the number of unattainable numbers?

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Some Challenges

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  • Challenge…A modification of the Chick McNugget Problem asks

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“What’s the largest unattainable order when boxes of 6, 9 and 20 are offered?”

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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
  • Thanks to our Villanova volunteers: Dr. Haymaker (director), VU students
  • Thank you all for coming out.
  • And a special thanks to our mentors!!!!

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Survey