Which number?

Dec 2019
52
1
ok
(I thought the answer was 5 but I could be wrong so who can help me?)

Santa Claus tells the three super-smart elves Alpha, Beta, and Gamma, “Each of you has received a card that carries a number from 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. Your three numbers are pairwise distinct, and the largest one equals the sum of the two smaller ones. Now, please have a look at your cards with your number, but don't show it to the two others!”

Alpha, Beta and Gamma stare at their cards for some time.

  1. After some time Alpha says, “According to my knowledge, there are at most eight possible candidates for Beta's number.”
  2. Then, Beta says, “According to my current knowledge, there are exactly three possible candidates for Gamma's number.”
  3. Gamma shouts, “I see! Now, I know Alpha's number.”
  4. Alpha thinks about it and says, “I still don't know Beta's number.”
  5. Beta exclaims, “I see! Now, I also do know Alpha's number.”


Of course, we would like to know: What is Alpha's number?
Calculations are required!
 

romsek

Math Team
Sep 2015
2,773
1,552
USA
This is not complex analysis. Please put these brain teasers in the General Math or Lounge forum.
 
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skipjack

Forum Staff
Dec 2006
21,321
2,386
Alpha''s number is 4. Hence Alpha initially knows that Beta has 1, 2, 3, 5, 6, 7, 9, or 10.
 

skipjack

Forum Staff
Dec 2006
21,321
2,386
One has to use the information given and make some reasonable assumptions. As Alpha knew that there were only eight possible candidates for Beta's number, Alpha's number was 4, 5, 8 or 10. As Beta, knowing Alpha's declaration, saw that there were exactly three possible candidates for Gamma's number, Beta's number must have been 4, 6, 7 or 9. Now try to continue reasoning in this way.
 
Last edited:
Dec 2019
52
1
ok
One has to use the information given and make some reasonable assumptions. As Alpha knew that there were only eight possible candidates for Beta's number, Alpha's number was 4, 5, 8 or 10. As Beta, knowing Alpha's declaration, saw that there were exactly three possible candidates for Gamma's number, Beta's number must have been 4, 6, 7 or 9. Now try to continue reasoning in this way.
How do you know Alpha's number was 4, 5, 8 or 10 tho? And why must Beta's number have been 4, 6, 7 or 9?
 

skipjack

Forum Staff
Dec 2006
21,321
2,386
If Alpha's number is 8, Alpha knows that Beta's number couldn't be 4, as that would leave no number available for Gamma. Do you understand that?