How many ways can I hold my girlfriends hand?
August 13, 2010 10:43 PM   Subscribe

How many different ways can you hold hands with someone?

I've got a nerdy girlfriend who loves math, and she's always the one impressing me with her math trivia...so I'd like to impress her for once!Her birthday is coming up and I want to write in her card how many different combinations of hand holding exist. (And how many consecutive days I will hold her hand without ever repeating blah blah blah :)

So if my fingers are (M) and her's are (H), standard, fully interlaced hand holding would look like:

MHMHMHMHMH and HMHMHMHMHM

and handshake-style hand holding would look like:

HMHHHHMMMM or MHMMMMHHHH

and a less common configuration might look like:

MMHMMHMHH


I would consider "hand holding" to be one or more of my fingers interconnected with at least one or more of her fingers. I realize a lot of the combinations won't be practical for hand holding, but if they can physically be done, they count!

Math people, please help!
posted by 777 to Science & Nature (12 answers total) 4 users marked this as a favorite
 
(I'm assuming that you don't want to consider holds where you do something like crossing your fingers, so you can either have two of your fingers between two of hers naturally or crossed)

For this specific case, there's 250 ways for you to hold hands. In general, there are n choose k - 2 ways for you to do this (where n is the total number of fingers and k is the number of fingers of you or your partner (it doesn't matter).

The way you can think of this is you have ten blank slots:

_ _ _ _ _ _ _ _ _ _

And you need to put your five fingers somewhere. She'll put her fingers in the other slots. The minus two comes from the two cases you want to ignore, all your fingers at the start of the hold or all of them at the end.

If you want, I can give you an enumeration of all the layouts of your hand holds, using your above notation.
posted by mge at 11:42 PM on August 13, 2010 [1 favorite]


I believe this is similar to asking "How many ways can you interleave two sets of cards in a riffle shuffle?", that is, so the ordering of the subsets are preserved in the larger sets.

Given m members in one set and n members in the other set, the result is (m+n)!/(m!n!). So in the case of 5 fingered hands, it'd be 10!/(5!5!) = 15120. Then, minus two for the degenerate methods, MMMMMHHHHH and HHHHHMMMMM, which isn't really hand holding at all.

So I'd say 15,118. Hopefully I didn't fudge anything, I'm not particularly good at combinatorics. Someone feel free to correct me if I'm off base. :)
posted by Herschel at 11:46 PM on August 13, 2010


My answer above is obviously way too big, because 10!/(5!5!) = 252 (not 15,000), minus 2 = 250, in line with mge. :)
posted by Herschel at 12:31 AM on August 14, 2010


The above posters are on the right track, but seem to be missing something your 'one or more of my fingers interconnected with at least one or more of her fingers' rule.

That is, every combination could be repeated but without your little fingers touching (though some combinations only differ in the last finger, so it would add a little less than 250 to the total). Then the same again but for the last two fingers not touching, all the way up until you're just hooking thumbs. Then you can start with every combination, but without the index or ring fingers touching, etc. If you only need two fingers to be connected to be considered a 'hold', then each MMHHMHMHHM combination has a range of possibilities within itself.

Then you could also start playing with things outside of the finger alignment set. Monkey grip is holding hands, but is distinctly different from MMHMHMHMHH (I suppose it would be M[MH][MH][MH][MH]MH). Then you can grasp each other's wrist, then you can hold her hand from above (your palm on the back of her hand) and various holds where some fingers are interlocked and others are grasping parts of the thumb or wrist.

It's a little too late for me to start playing with numbers (and I doubt I'd remember enough to do it anyway), but it seems there are more options beyond the 250. No matter though, you can record this in your birthday card and then play together with more and more obscure ways of holding hands!
posted by twirlypen at 12:56 AM on August 14, 2010 [1 favorite]


Are you planning to consider practical factors also, or just simple combinatorics?

For example, MHMMHHHHMM is a valid combination according to the rules presented so far (I think), but you wouldn't be able to do it unless your hand is drastically bigger than hers.

You could try refining your algorithm to exclude such impossible combinations (e.g. more than three fingers from the same hand in a row), in which case an enumerative approach would probably be simpler to code.
posted by Dr Dracator at 2:13 AM on August 14, 2010 [1 favorite]


I suppose if you wanted to take this even further, you could look into holding her left in yous left (across your body), her right in yours (hers behind her body) and so forth.
posted by twirlypen at 2:20 AM on August 14, 2010


The above posters are on the right track, but seem to be missing something your 'one or more of my fingers interconnected with at least one or more of her fingers' rule.

That's accounted for by the subtraction of two; you follow mge's logic and find there's 252 combinations then you subtract two, because of there are two invalid combinations - namely MMMMMHHHHH and HHHHHMMMMM. So, 250.
posted by Mike1024 at 2:38 AM on August 14, 2010


Yes, those two don't count, but every other combination assumes all ten fingers are interlocked. MHMHMMHHMH counts as one combination out of the 252, but MHMHMMHH with the last two fingers free is another unique combination.
posted by twirlypen at 3:44 AM on August 14, 2010


Can't you just do the same thing but with three options (each of the ten spaces could be blank, subtracting a couple that have none of her fingers or none of his?)

I am terrible at math.
posted by SMPA at 5:46 AM on August 14, 2010


I'm not sure if this will count by your definition, and my wife certainly does not accept this as regulation hand holding... but if you would like to be really irritating, grasp her pinky in your thumb and forefinger. Many other similar combinations possible.
posted by Meatbomb at 7:20 AM on August 14, 2010


I don't know how this would factor in or if it's accounted for in the above equations, but I sometimes hold hands by grabbing my boyfriend's index finger with my whole hand, which I guess would be:

HMHMMMMMHHHHH

I figure the "grabbing" technique could maybe account for some of the suggested "impossible" combinations.

I've also held hands in the top-over-bottom, grasping way, which works best laying down. It's hard to describe this one. You can do it easily in front of you by "locking" your hands together where your arms are making a horizontal line, one thumb pointing up, the other pointing down. Hope that makes sense.
posted by a.steele at 8:15 AM on August 14, 2010


Response by poster: twirlypen: this is exactly what I'm looking for...I want to include all combinations that also include less than ten fingers, as well as holding her hand from above or below, etc. Thanks!
posted by 777 at 9:29 AM on August 14, 2010


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