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### Re: Generating all 5-card hands

by hv (Parson)
 on May 19, 2005 at 12:12 UTC ( #458597=note: print w/replies, xml ) Need Help??

in reply to Generating all 5-card hands

For a hand with a 5 to avoid scoring 2 points, it must have no pair, and no subset that sums to 15. Since that means we can only have one of each of the pairs (1, 9), (2, 8), (3, 7), (4, 6) that sum to 10, the hands to consider consist of the 16 ways we can choose one from each of those pairs.

If we have 9 we cannot have 6, so we must have 4, so we cannot have 2, so we must have 8, so we cannot have 7, so we must have 3. But 3+4+8 = 15, so we cannot have 9.

So we must have 1; if we have 8 we cannot have 7, or 6, so we must have 3 and 4. So we have 3+4+8 = 15 again, which means we cannot have 8.

So we must have 1 and 2, which means we cannot have 7, so we must have 3. But 1+2+3+5+4 = 15, and 1+3+5+6 = 15. So there is no hand including a five which doesn't include at least 2 points.

Hugo

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Re^2: Generating all 5-card hands
by Anonymous Monk on May 05, 2008 at 22:46 UTC
I thought you would be interested to know that I added a link to this discussion at Cribbage Corner, in the section on hand distribution and probabilities. Thank you for taking the trouble to work this out!

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 [LanX]: two utf8 strings from different sources are base64 encoded, but after joining both the umlauts in teh second get deleted [Corion]: LanX: You can't just join two base64 strings together [LanX]: (not a high priority bug because I can use some HTML entities in the second string) [Corion]: base64 is padded to a multiple of 4 chars (or something) [LanX]: misunderstanding, I joined them before converting to base64 [Corion]: Also, I would be wary of encodings and try to make really sure that both input strings are UTF-8. Maybe join the input strings from one source together to see whether they decode as bad or not [Corion]: LanX: Then the problem should persist without encoding to base64 too ;) [LanX]: I think it's a flag problem ... I'll produce a reprodocable example for SOPW [Corion]: "flag problem" to me sounds like "contains UTF-8 bytes but was never properly decoded to an UTF-8 string"

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