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?
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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:
0.5,1.5,2.5,3.5,4.5,5.5,6.5,1.5,3.5
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. 🙂
Sorry!!did a calculation mistake.And by integers, here I mean distinct integers.