Math Detours · Grade 7 · Around the corner · 13

I cut, you choose

Two people can share a cake so neither envies the other. Three is harder, and the method shows what "fair" means.

Name ____________________Date ____________

Tonight's idea

Two people share a cake. Each wants at least half by their own taste. Tastes differ, so a ruler will not settle it.

One person cuts the cake into two pieces. The other person chooses.

The cutter cuts so that she would be happy with either piece. Whichever piece is left, she has half by her taste. The chooser takes the piece he likes more, so he has at least half by his.

chocolate vanilla Ann cuts at 4.5 cm 06 cm12 cm Ann9 points9 points Bob4.5 points7.5 points
One cut, scored two ways. Ann sees two equal pieces. Bob does not.

This cake is 12 cm long, half chocolate, half vanilla. Ann scores chocolate at 2 points per cm and vanilla at 1: \(12 + 6 = 18\) points in all. Bob scores every cm as 1 point: 12 in all. Ann cuts at 4.5 cm, because \(4.5 \times 2 = 9\), half of 18. Bob takes the longer piece. Ann has \(\tfrac{9}{18}\), exactly half. Bob has \(\tfrac{7.5}{12}\), more than half.

That is what fair means here: each person gets at least an equal share by their own measure.

Watch one: three people

Ann, Bob and Cy each want at least a third.

  1. Ann cuts a piece she thinks is exactly a third.
  2. Bob may trim it, if he thinks it is more than a third. The trimmings go back with the rest of the cake.
  3. Cy may trim it too.
  4. The last person to cut or trim the piece takes it and leaves.
  5. The other two share the rest by "I cut, you choose".

Why it works: the taker thinks the piece is exactly a third. Each of the others thinks it is a third or less, or they would have trimmed it. So they think at least two thirds is left, and half of that is a third.

Your turn

  1. Same cake, but now Bob cuts and Ann chooses. (a) Where does Bob cut? (b) Which piece does Ann take, and what fraction of the cake is it worth to her? (c) Would Ann rather cut or choose?
  2. Ann cuts at 6 cm, hoping Bob will leave her the chocolate. (a) What is each piece worth to Bob? (b) Bob takes the chocolate. What fraction of the cake does Ann get, by her measure? (c) Why is 4.5 cm her safe cut?
  3. Three people, a new cake. Ann cuts a piece she thinks is a third. Bob thinks it is worth 40% of the cake. Cy thinks it is worth 30%. (a) Who trims it? (b) Who takes it? (c) Cy thinks the cake that is left is worth at least what percent?
  4. Carry on from question 3. Ann and Cy share what is left by "I cut, you choose". (a) Explain why Cy ends with at least 35% of the whole cake, by his measure. (b) Explain why Ann ends with at least a third, by hers.

Stretch: fair is not the same as happy

In question 3, Bob left with a piece he thinks is exactly a third. Then he watches Ann and Cy share the rest. Could Bob think Cy ended up with more than he did? Give percentages, by Bob's measure, that show how. Was the promise to Bob broken?

Answers

  1. (a) At 6 cm: each piece is worth 6 points to him. (b) The chocolate half. It is worth \(6 \times 2 = 12\) of her 18 points, which is \(\tfrac23\). (c) Choose: \(\tfrac23\) beats the \(\tfrac12\) she gets as cutter. The cutter is only sure of exactly half.
  2. (a) 6 points each, so he has no reason to pick the vanilla. (b) The vanilla is worth 6 of her 18 points: \(\tfrac13\). (c) At 4.5 cm both pieces are worth 9 to her, so whatever Bob does she gets half. Any other cut makes one piece worth less than half to her, and she might be left with it.
  3. (a) Bob, because 40% is more than a third (about 33.3%). He trims it down to what he thinks is a third. Cy does not: he thought it was 30% before it shrank. (b) Bob, the last to trim. (c) At least 70%, since he thinks the piece taken is worth 30% or less.
  4. (a) Cy thinks at least 70% is left. If he cuts, he makes two pieces he thinks are equal, each at least 35%. If he chooses, he takes the piece he thinks is bigger, at least half of it. (b) Ann thought her piece was exactly a third, and Bob did not make it bigger. So she thinks at least two thirds is left, and by the same argument she gets at least half of that.

Stretch. Yes. Bob thinks the rest is worth two thirds, about 66.7%. Ann and Cy share it by their tastes, not his. By Bob's measure the two pieces might be 26.7% and 40%, with Cy holding the 40%. Bob has his third, so the promise is kept, but he envies Cy. With two people this cannot happen: if you have at least half, the other person has at most half.

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