Puzzle #135: Two Dice = Nine Cards

This is a gem of a puzzle from Gurmeet.net, one of my favourite collection of puzzles!

Alice has two standard dice with labels 1 thru 6. When she rolls them and adds their labels, she gets a distribution over integers in [2, 12]. Bob has nine cards, each labeled with some real number. When Bob chooses two cards (without replacement) and adds their labels, he gets exactly the same distribution over integers in [2, 12] as Alice gets by rolling her dice. What are the labels on Bob’s nine cards?

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 dicey week to you all!

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3 Responses to Puzzle #135: Two Dice = Nine Cards

  1. Abhinav Jain says:

    Clearly if any one of the labels is an integer than all other have to be integer.Now all labels can’t be integers since there will always exist a pair which has sum greater than 12 or less than two.now the sum of fractional part of any two labels should be 1.Now we can solve for them and all of them will be equal to 0.5.Now with a bit of hit and trial we can figure out the labels:

  2. Girish Tutakne says:

    Abhinav – the last 2 should be 2.5 and 4.5 (instead of 1.5 and 3.5): so the sequence is 0.5, 1.5, 2.5, 3.5, 4.5, 5.5, 6.5, 2.5 and 4.5. The logic about integers is not really sound – because we can have 9 numbers to always have the sum totals fall within the 2 and 12 integer range (eg: 1, 1, 2, 3, 3, 4, 5, 6, 6 almost gets the right distribution but not quite with all possible totals being the numbers from 2 to 12).
    Still trying to figure out if there is a less than brute force version for this. I have done it with an Excel table, but it is not elegant. 🙂

  3. Abhinav Jain says:

    Sorry!!did a calculation mistake.And by integers, here I mean distinct integers.

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