• ".uoy тззm oт bзzaзl9”

# The number of people who have shaken hands an odd number of times is even

Inspired by Pisco Bandito's excellent implementation, Prozac74's mirror set, and to some degree, Lewis Carroll's Through the Looking Glass

The statement of the title is true, don't listen to what some other photos might tell you. Below are two proofs.

Proof by Mathematical Induction
We induct on the number of times people have shaken hands.
Let E be the set of people who have shaken hands an even number of times.
Let O be the set of people who have shaken hands an odd number of times.

Base Case
Before a handshake ever took place, O has size zero and is therefore even.

Induction Step
We assume the statement is true for k handshakes. We need to prove the statement for k+1 handshakes.

For the (k+1)th handshake, one of the following cases is true:
A. Two people from E shake hands (who now move to the set O).
B. Two people from O shake hands (who now move to the set E).
C. One person from O shakes hands with one person from E (they now "trade membership")

For each case, the parity of O remains the same. And since the size of O was even beforehand (by our induction hypothesis), the size of O remains even.

Seb Przd's Proof Using a Parity Argument
O and E are defined as above.
Let n be the number of times any hand has shaken another.
Let n_o be the number of times a hand from the set O has shaken another.
Let n_e be the number of times a hand from the set E has shaken another.

Note: n = n_o + n_e

Each handshake increases n by two, hence n is even. n_e is a sum of even numbers (by definition) and hence is also even. Since both n and n_e are both even, the equation above tells us that n_o is even as well. But n_o is a sum of odd numbers (by definition). The only way for n_o to be even then, is if the size of O is even.

Éireannach Súil, bacimaging, and 311 other people added this photo to their favorites.

1. now this is tight...

2. this is very very funny

3. so peculiar!!!

Join U-MANI group!

4. This is sooo cool. Well done!

5. so clever!

6. Hi, I'm an admin for a group called Anything Absurd, and we'd love to have this added to the group!

7. Panzimar [deleted] 53 months ago

Clever and so well done. The backward type is an excellent detail. :)

8. holy hell!!

i love this!!

9. Please don't mind the fave! I want to learn how to do this!

10. Haha cool that's so odd!

11. Very cool!

12. Excellent

13. Oh, it's so nice!!!!

14. Pete, hanyi, thanks for commenting.

15. Talking about hands... you may have to much time on them. :-)

But seriously, good job on the picture.

16. Hey thanks Carlos, I appreciate the comment, even after all this time.

I suppose some people watch Dancing with the Stars result episodes, some people play call of duty, and some people take up photo editing as a hobby. ;)

I'm curious though, how much time you think is spent on a photo like this? This one took not even an hour.

cheers

17. That´s true. It´s fun to play with images like that.

18. Wow, this is pretty impressive and hard to do...it looks like the reflection is almost popping out. I have a similar shot using the mirror reflection as a multiplicity trick but no where near as good as your. Cheers

[ via tag search on flickriver  --tjunkie (Art is never finished, only abandoned) ]

19. Noticed you guys use different products there on the counter...And
apparently the closest twin would have musical note for a towel monogram?

20. Lionhearted: yeah I prefer bars of soap but what can ya do.

The forwards S is merely because the photo is actually mirrored left to right: I thought the photo would look better that way compositionally. and I thought the mirrored S (for Swart) looking proper was a neat side effect so I didn't uncorrect it.

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