Fingers Crossed, You'll Get It Right!
For the purposes of this problem, each thumb should be considered to be just another finger.
I am dextrous enough that I can cross any adjacent pair of fingers on either hand, but not dextrous enough to be able to cross non-adjacent fingers. This means that at any one time, I can cross at most two pairs of adjacent fingers on each hand.
If I cross fingers in this way on either or both hands and then touch my fingers together (with palms facing each other) so that all 5 fingers on my left hand are paired with a finger on my right hand, in how many ways can my fingers ultimately be paired? Submit this.
Clarifications
Note that some combinations of crossings will yield the same result. A couple of examples should help to clarify:
-
If I cross pinkie and ring fingers and also thumb and index fingers on both hands the combination of touching fingers is the same as if I left all fingers uncrossed. (So only one is added to the count when both of these cases are considered.)
-
If I cross middle and index fingers on my left hand while leaving the fingers on my right hand uncrossed, that will result in the same fingers touching as if I left the fingers on my left hand uncrossed and crossed the middle and index fingers on my right hand. (So, again, only one is added to the count when both of these cases are considered.)
Happy finger-crossing!
The answer is 24.
This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try
refreshing the page, (b) enabling javascript if it is disabled on your browser and,
finally, (c)
loading the
non-javascript version of this page
. We're sorry about the hassle.
1 solution
If you don't cross any finger on your left, the way you can cross your fingers on your right is
These pairings don't include (1,3) (2, 4) (3, 5). We will consider these cases separately.
For (1, 3), left hand must cross (1, 2) while right hand cross (2, 3). There are 3 configurations ((3, 4, 5), (4, 3, 5), (3, 5, 4)) on your left hand to do that while 2 for your right hand. Hence the number of pairings are 3 × 2 = 6 .
Similarly, (2, 4) and (3, 5) have 4 and 6 pairings respectively.
Notice that they are mutually exclusive, you cannot have (1, 3) and (2, 4) together. Hence we need not to consider overlappings.
The total number of pairings is 8 + 6 + 4 + 6 = 2 4 .