The Pigeonhole Principle · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

More pigeons than holes forces a repeat

Mathematics · Discrete Mathematics · ages 19-20
Name ______________________   Date ____________

  1. Ten pigeons roost in 3 holes. At least how many share one hole?

    Answer: ______________

  2. Four socks are drawn from a drawer with 3 colours. What must happen?

    • all four differ
    • each colour appears at least once
    • three share a colour
    • two share a colour

  3. Four socks are drawn from a drawer with 3 colours. What must happen?

    • Each colour appears at least once
    • All four differ
    • Two share a colour

  4. In any group of 13 people, two share a birth month. True or false?

    Circle one:   True   False

  5. Four holes hold pigeons. How many pigeons force some hole to contain at least 3?

    Answer: ______________

  6. Seven integers are chosen from 1 to 12. Why must two of them sum to 13?

    • 12 is an even number
    • The six complementary pairs summing to 13
    • 7 is a prime number

  7. In a group of 5, two people share a handshake count. The holes in the argument are what?

    • the 5 people
    • the 4 possible handshake counts 0, 1, 2, 3
    • the 10 handshakes
    • the 5 counts 0 to 4

  8. Generalised form: m pigeons in n holes force some hole to hold at least m/n rounded up. True or false?

    Circle one:   True   False

  9. A drawer holds socks in 5 colours. How many socks guarantee two of the same colour?

    • 5
    • 10
    • 6

  10. A drawer holds socks in 5 colours. How many socks guarantee two of the same colour?

    • 5
    • 10
    • 6
    • 11
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Answer key

For grown-ups. Fold this page away before handing over the rest.

More pigeons than holes forces a repeat W1-mt_wgfpmZqzd9-s1

  1. 4 · 4. Since 10/3 exceeds 3, some hole holds 4 or more.
  2. two share a colour · Two share a colour. With more socks than colours, some colour repeats.
  3. Two share a colour · Four socks into three colours overfills the colours, so some colour repeats.
  4. True · Twelve months are the holes and thirteen people overfill them.
  5. 9 · 9. Two per hole absorbs 8 pigeons, so the ninth forces a triple.
  6. The six complementary pairs summing to 13 · Pairs (1, 12) through (6, 7) host 7 chosen numbers, so one pair gives both members.
  7. the 4 possible handshake counts 0, 1, 2, 3 · The 4 possible handshake counts 0, 1, 2, 3. Counts 0 and 4 exclude each other, leaving 4 live holes for 5 people.
  8. True · True. Otherwise every hole below the ceiling totals under m pigeons.
  9. 6 · Five socks can show each colour once, but the sixth repeats one.
  10. 6 · 6. Five socks can show each colour once, but the sixth repeats one.
Worksheet · LightMySky