Puzzle #169: Division of a Cake

This is a very old puzzle. I remember this puzzle being asked by my Chemistry teacher in Class XI – Dr. Ravi Gopinath, someone was has had a strong influence in my passion for puzzles. I got reminded of this puzzle while having breakfast yesterday with my kids and one of their friends.

Anisha and Arushi need to divide a cake (any shape) between themselves and want the division to be fair, then the process is very simple. One person (say Anisha) divides the cake into two parts which she thinks are equal, and Arushi can pick one. Neither party has a reason to be unhappy and think that it was a fair division. Now they have a friends of their – Nishtha – visiting. They need to divide a red velvet pancake into 3 parts so that each one of them thinks that the division is fair. How do they do it?

What happens when their mom and dad also want a fair share, so there are 5 now? n?

As always, please send your answers as comments within the blog (preferred), or send an e-mail to alokgoyal_2001@yahoo.com. Please do share the puzzle with others if you like, and please also send puzzles that you have come across that you think I can share in this blog.

Happy divisions!


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7 Responses to Puzzle #169: Division of a Cake

  1. Prakhar says:

    Not sure if my response got posted before. Along the lines of 2 , for three , let the first one cut the cake and keep aside 1/3. Now the second person divides the remain and let’s the third person pick the one piece from the three created. Then the second person picks from the remaining and the first person gets in the end. Similarly for n people, person cuts 1/n out, person 2 makes 1/n-1 fraction piece out, and so on. The last person picks first and we move backwards.

    • Suman says:

      I don’t think this is correct. For fair or envy free division, I should be unhappy only if I made a mistake (either in cutting or choosing). In your example, if person 2 makes a bad cut and lets assume splits the 2/3rd into 40:60 instead of 50:50, person 1 suffers due to his mistake. Person 3 will pick up 60% of the 2/3rd. person 2 would realize his mistake and pick up the first piece. person 1 would be left with 40% of the bigger piece for no fault of his!

  2. For n=3,
    The first man cuts the cake into 3 pieces. The second man picks the largest piece and divides it into two such that the two largest pieces amongst the 4 pieces created are equal. The third man picks the largest piece and keeps it for himself. The second man picks the largest post that. The first person keeps the rest.

    • Suman says:

      I am not sure this is correct either. Please see my comment above.

      let’s say the first man cut it into
      27:27:46 (instead of 33.33:33.33:33.33) 🙂 just a dramatic example
      the second man tried to fix it and and chopped the 46 piece
      27:27:28:16 (trimmed piece)
      the 3rd man in the best case is left with 28. Even if the second man made the perfect fix and made it
      27:27:33.33:15.33, the 3rd man gets his fair share, but the second man is hosed.

  3. Suman says:

    so stupid of me with my fat fingers. please read the last one as:
    sorry about that. i was thinking of other numbers in my head while typing this and it lead to the mess. But the idea remains the same 🙂

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