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Mathematics LibreTexts

5.6: Moving Knife

  • Page ID
    34202
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    A somewhat different approach to continuous fair division is called the moving knife procedure.

    Moving Knife Method

    In this method, applied to a cake,

    1. A referee starts moving a knife from left to right across a cake.
    2. As soon as any player feels the piece to the left of the knife is worth a fair share, they shout “STOP.” The referee then cuts the cake at the current knife position and the player who called stop gets the piece to the left of the knife.
    3. This procedure continues until there is only one player left. The player left gets the remaining cake.

    Example 10

    Suppose that our four salespeople from above decided to use this approach to divide Washington. Rather than move the “knife” from left to right, they decide to move it from top to bottom.

    Solution

    A map of the state of Washington with a horizontal line drawn about a quarter of the way from the top, with the area above shaded pink.The referee starts moving a line down a map of the state. Henry is the first to call STOP when the knife is at the position shown, giving him the portion of the state above the line.

    A map of the state of Washington with a horizontal line drawn about a quarter of the way from the top, with the area above shaded pink, and a second horizontal line drawn about halfway from the top, with the area between it and the first line shaded orange.Marjo is the next to call STOP when the knife is at the position shown, giving her the second portion of the state.

    A map of the state of Washington with a horizontal lines drawn about one quarter, one half, and three quarters from the top, with the four segments shaded pink, orange, yellow, and blue.Bob is the next to call STOP, leaving Beth with the southernmost portion of the state

    While this method guarantees a fair division, it clearly results in some potentially silly divisions in a case like this. The method is probably better suited to situations like dividing an actual cake.


    5.6: Moving Knife is shared under a CC BY-SA 3.0 license and was authored, remixed, and/or curated by David Lippman via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

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