==> induction/handshake.s <==
Assume there were 2n people (including host and hostess)
in the party.
1. When the host asked the question he must have got
2n-1 responses (including from his wife).
2. All of the responses were different.
The responses have to be (0, 1, 2, 3, ..., 2n-2)
to satisfy the above requirements. As 2n-2 is the maximum
possible handshakes any person in this party could have made.
/** Below,
P{x} - means a person who shook x hands.
H - means the host
**/
H: <-------->2n-2 0
2n-3 1
2n-4 2
2n-5 3
. .
. .
. .
n n-1 n-2
(There are 2n-1 on the circle.)
P{2n-2} must have handshaked with H (because in the circle he
can handshake with only 2n-3. He has to exclude himself also.)
P{2n-3} must have handshaked with H (because in the circle he
can handshake with only 2n-4.)
P{2n-4} must have handshaked with H (because in the circle he
can handshake with only 2n-5.)
P{n} must have handshaked with H (because in the circle he
can handshake with only n-1.)
from P{n-1} to P{0} nobody handshakes with H, because, for them
the handshake numbers are satisfied on the circle itself.
This leaves H with (n-1) handshakes.
----------------------------------
P{0} must be the spouse of P{2n-2} (since P{2n-2} handshakes with
everbody else.)
.
.
.
same logic till P{n-2}
leaving the Hostess to be P{n-1}.
----------------------------------------
So,
Host - made (n-1) handshakes.
Hostess - made (n-1) handshakes.
where n is the number of couple including
the host couple.
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