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Three Cuts Through a Cake (illustration)
FILE No.95Unsolved

Three Cuts Through a Cake

Difficulty ★★☆about 5 minLogic PuzzleCuttingIntersections
Three straight cuts through a round cake. What is the most pieces you can get?

Case Record

You want to cut a round cake into as many pieces as possible with three straight cuts of a knife. You do not move the pieces or stack them.

Each cut goes straight down through the cake, from one edge to the other, so seen from above you are drawing three straight lines across a circle. The cuts may be placed anywhere.

One cut makes 2 pieces. Two cuts make at most 4.

Your question

What is the largest number of pieces three cuts can make?

Case Files

Evidence

Diagram

Maximum piecesCutsPieces01122437The gain grows: +1, +2, +3.
Diagram

Timeline

  1. You want as many slices as possible.

Hints

If you get stuck, open the hints one at a time, in order.

Open hint 1

How many pieces do 2 cuts make at most?

Open hint 2

How should the third cut cross the others to add the most?

Open hint 3

Into how many segments is the third cut divided?

Solve It

Your question

What is the largest number of pieces three cuts can make?

Choose the answer closest to the truth.

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The Truth

7

The answer is: 7.

Make the third cut cross both of the first two cuts inside the cake. It crosses two lines, so it is split into 3 segments, and each segment splits one piece in two. That adds 3 pieces: 4 + 3 = 7.

If all three cuts pass through the same point, you only get 6.

Cross the other twoCross both earlier cuts inside the cakeThe third cut splits into 3 segmentsEach adds a piece: 4 + 3 = 7
Diagram of the solution

Key Points

  • Make each cut cross all the earlier cuts
  • New pieces = crossings + 1
  • Cuts through one point add fewer

Why people get fooled

Cutting through one point, or parallel cuts, gives fewer pieces.

Similar tricks

  • The intersection-counting type
  • The small-cases type

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