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?
This puzzle is from https://gurmeet.net/puzzles/two-dice-nine-cards/index.html
My approach: Sum of 2 die faces should be 1, so this suggests all's fractional part would be 0.5, now, 5.5+6.5 gives 12, and 0.5+1.5 gives 2, so we can safely say 0.5,1.5,2.5,3.5,4.5,5.5,6.5, these would be there and now, expected value of the dice should be same as 3.5 thus every frequency is getting multiplied by 2 here, for eg 1+1 = 2, but (0.5,1.5) and (1.5,0.5) gives 2, thus for 3, we have 1+2, 2+1, so here we should have 4, thus 0.5+2.5, 2.5+0.5, none thus, add 2.5 to this to get 4, and similarly added 4.5 too, by symmetry. Thus the total values are 0.5,1.5,2.5,2.5,3.5,4.5,4.5,5.5,6.5. However, how did we rule out that there cannot be integers here in nine cards?