Blue-haired wine math puzzle (solution)

Be sure to try the blue-haired puzzle first.

The answer is that you can test all the cases of wine, positively identifying the one that will turn your hair blue.

(a little spoiler space)

In fact, you can test up to 243 cases this way.

This puzzle can be solved by working backward from how many cases of wine you’ll be able to test on the second day, based on how many dukes’ hair turns blue the first night. For example, if no dukes’ hair turned blue during the first night, on the second night you’ll be able to test 32 cases of wine conclusively, because you still have 5 dukes and 2^5 = 32. I.E. an arrangement like this, with dukes A,B,C,D, and E:

Case    A         B         C         D         E

1     drinks    drinks    drinks    drinks    drinks

2               drinks    drinks    drinks    drinks

3     drinks              drinks    drinks    drinks

4                         drinks    drinks    drinks

5     drinks    drinks              drinks    drinks

6               drinks              drinks    drinks

7     drinks                        drinks    drinks

8                                   drinks    drinks

9     drinks    drinks    drinks              drinks

10              drinks    drinks              drinks

11    drinks              drinks              drinks

12                        drinks              drinks

13    drinks    drinks                        drinks

14              drinks                        drinks

15    drinks                                  drinks

16                                            drinks

17    drinks    drinks    drinks    drinks      

18              drinks    drinks    drinks      

19    drinks              drinks    drinks      

20                        drinks    drinks      

21    drinks    drinks              drinks      

22              drinks              drinks      

23    drinks                        drinks      

24                                  drinks      

25    drinks    drinks    drinks                  

26              drinks    drinks                  

27    drinks              drinks                  

28                        drinks                  

29    drinks    drinks                              

30              drinks                              

31    drinks                                          

32                                                    

So working backwards, on the first day, there can be 32 cases of wine that no duke drinks, because then if no one turns blue, you have the above situation and you can nail down exactly which case is tainted on the second day.

If one duke turns blue on the first day, you can distinguish 16 wines on the second day: 2^4 is 16, and you have cases 1-16 above without E. So on day one, each duke should drink 16 wines that no other duke drinks — 5 dukes * 16 cases = 80 cases tested by exactly one duke each on day one.

Likewise, if two dukes turn blue on day one, you can distinguish 8 wines on day two: 2^3 = 8, and you have cases 1-8 above with just dukes A, B, and C.

There are 10 ways to pick two dukes out of 5: 10C2 = 5 * 4 / 2. So you have 10 combinations, drinking 8 wines each that no other pair of dukes drink, or 80 cases tested by exactly two dukes each on day one.

If three dukes turn blue on day one, you can distinguish 4 wines on day two: 2^2 = 4, and you have cases 1-4 above with just dukes A, and B.

There are 10 ways to pick three dukes out of 5: 10C3 = 5 * 4 / 2. So you have 10 combinations, drinking 4 wines each, or 40 cases tested by exactly three dukes each on day one.

If four dukes turn blue on day one, you can distinguish 2 wines on day two: 2^1 = 2, and you have cases 1-2 above with just duke A.

There are 5 ways to pick four dukes out of 5. So you have 5 combinations, drinking 2 wines each, or 10 cases tested by exactly four dukes each on day one.

Finally, you can have all five dukes share one wine on day one. If they all turn blue, that’s the case.

So on day one you have:

  • 32 cases that no duke tests
  • 80 cases where each case is tested by exactly one duke
  • 80 cases where each case is tested by exactly two dukes
  • 40 cases where each case is tested by exactly three dukes
  • 10 cases where each case is tested by exactly four dukes
  • 1 case that that is tested by all five dukes

Total: 243

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One thought on “Blue-haired wine math puzzle (solution)

  1. Pingback: The blue-haired wine math puzzle | Geoff Canyon's Appeal to Authority

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