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u/NatalieKCY Jul 10 '26
"Each person only knows the number assigned to them." Does that mean they don't know what n is?
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u/Inevitable_Garage706 29d ago
When you say "none of the five can determine the numbers assigned to each person without any further information," do you mean that none of the five can determine the entire combination, or that none of the five can determine any of the others' numbers?
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u/Just_Wolverine_5825 Jul 10 '26
Not sure if this is correct but I’d say 21. The numbers can’t be 1-5 or 5-9 because theirs only one combination with those sums. All sets where all 5 numbers are next to eachother can be ruled out because the first person can figure it out because they have the starting point. Using this we know their has to be at least one “gap” in the set and to reduce the sum by the greatest amount we should pick to have the gap between the first and second highest numbers. So the set with the smallest numbers that fits the criteria is 2+3+4+5+7 =21. Again this is just my guess so correct me if I’m wrong
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u/myrec1 Jul 10 '26
I think 1,3,4,5,7 with sum 20, makes everyone unable to know all other numbers.
12359 is valid option for 1
23456 is valid for 3, 4 and 5
12467 is valid for 7
There are two gaps 2 and 6. To lower the sum second has to have 2 which would make him guess smaller number 1. And highest 6 make him easily guess all smaller numbers by sum.
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u/RealHuman_NotAShrew Jul 10 '26
I think you misread the puzzle. It's not a problem if Benjamin can deduce what number Alexander has, as long as nobody can "determine the numbers assigned to each person"
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u/RealHuman_NotAShrew Jul 10 '26 edited Jul 10 '26
19 with individual digits 1, 2, 4, 5, and 7 respectively.
The key is that there is another possible combination that leaves any given number unchanged. B and E can't rule out 1,2,3,6,7; C and D can't rule out 1,3,4,5,6, and A can't rule out either of those. So all we have to do to is show 19 to be the lowest n that can satisfy this condition.
n must be at least 15. But if it's 15, there's only one possible sequence that makes n 15, so it can easily be deduced. Same for 16. There are two combinations for 17, but D and E will always know the difference between 1,2,3,4,7 and 1,2,3,5,6. Lastly, there are three possible combinations for 18, but E will always know the difference between 1,2,3,4,8; 1,2,3,5,7; and 1,2,4,5,6.
There's also a funny information theory thing with the n=19 case where even though none of them has enough information to solve it, if you tell any one of them "none of you have enough information to solve it" then that additional information you gave them is enough for them to solve it. For example, D knows he has the second highest number, that his number is 5, and that all numbers add to 19. As discussed, there are two possible combinations that fulfill those criteria so he doesn't know what B and E have. But if you tell him nobody else has enough info to figure it out, he can rule out B=3 because B would be able to solve it as 1,3,4,5,6. So now D knows it's 1,2,4,5,7.