Comments on: The Sum of Some Hats

By: jd2718

jd2718 — Tue, 03 Oct 2006 03:22:11 +0000

Review, and fill in, correcting error. Discovering mistake.

Alan says no. Eliminates 2:1:1 (corrects mistake above)
Bert says no. Eliminates 1:2:1. 2:3:1 (corrects mistake above)
Graham says no. Eliminates 1:1:2. 2:3:5. 1:2:3
Alan says no. Eliminates 3:2:1. 4:3:1. 3:1:2. 8:3:5. 5:2:3

and I stand corrected and chagrined. I altered a number and made multiple answers possible.
Bert might see 3:B:1 and conclude 3:4:1
Bert might see 4:B:1 and conclude 4:5:1
Bert might see 3:B:2 and conclude 3:5:2
Bert might see 8:B:5 and conclude 8:13:5
Bert might see 5:B:3 and conclude 5:8:3

The first and fourth possibilities yield solutions: 39:52:13 (your answer) or 32:52:20 (my intent)

Apology? Of course. I goofed.

Fixable? Sure. Let Bert conclude his hat is 78.

By: jd2718

jd2718 — Sat, 30 Sep 2006 22:02:28 +0000

OK, let’s go further. When Graham says he doesn’t know, he provides us with a ton of information.
If he saw 1:2:G, he knows 1:2:1 is out, so he would have known. 1:2:3 is out.
If he saw 1:3:G he would have known, so 1:3:4 is out.
Also, 1:1:2 is out (he would have known instantly)

I don’t think I’ve skipped any.

Anyone else ready to jump in?

By: jd2718

jd2718 — Fri, 29 Sep 2006 00:55:43 +0000

Here’s a start.

When Alan says he doesn’t know, all three understand that he doesn’t see equal numbers on Bert and Graham’s hats. The ratio 1:1:2 is not correct, and all three know it.

When Bert says he doesn’t know, all three understand that the ratio 1:2:1 is out. Further, all know that Bert did not see 1:x:2, since he would have understood that the ratio was 1:3:2, and would have answered. So 1:3:2 is out.

By: Leigh Hunt

Leigh Hunt — Thu, 28 Sep 2006 19:08:48 +0000

Would you mind posting your answer to this one, Jonathan?

By: jd2718

jd2718 — Fri, 22 Sep 2006 22:30:16 +0000

Sorry so slow to respond. It appears that Alan would get his number on the second pass if it were 13 52 39.

I need to look more closely though, because I may have carelessly modified the problem and generated multiple answers. Give me through the weekend to reanalyze.

By: shuusaku

shuusaku — Thu, 21 Sep 2006 06:24:04 +0000

I got 13 from the sequence 52,26,13 and started out testing some combinations. This sequence is interesting as combinations where two have the same number are very limiting.

By: jd2718

jd2718 — Thu, 21 Sep 2006 00:38:16 +0000

There is something wrong, but I am not having an easy time finding it. Where did you find the numbers 13 and 39?

By: Shuusaku

Shuusaku — Wed, 20 Sep 2006 16:14:06 +0000

Alan 13
Bert 52
Graham 39

Reasoning goes like this:

Alan sees 52 and 39 he knows his number is 13 or 91
Bert sees 13 and 39 he knows his number is 26 or 52
Graham sees 13 and 52 he knows his number is 39 or 65

Alan can’t say his number
Bert can’t say his number
When Graham can’t say his number Bert thinks,

If I am number 26 Graham would see 26 and 13 and conclude that he was number 39 as 13 would be impossible(otherwise I would see two number 13 and be able to answer).
As Graham can’t answer I can’t be number 26 so I must be number 52.

Alan can’t still know his number on his second turn.