The math-hunters at grahamkendall.com have a lively
discussion, including this excerpt:
Rediscovery is not a silly issue. How many re-discoveries
have you made? I would guess many.
As youths (16) we played with the 5 cent 25 hole pinball
machines in the pool room. These paid off 5 cents per game that you won.
When you decided to cash in your games which of course was a rare event, we
called the clerk Gillie, who had bad eyesight, to come over and verify the
counter, flip the reset switch and pay us the money which of course went
right back into the machine. When we won, Gillie would come over, squint,
and look at the counter. Then he would tell us to flip the switch. In doing
so we noticed that the switch could be toggled back and forth quickly thus
stopping the won game count down. So if we had 100 games, we could quickly
flip the switch on and off to stop the counter at 99. Then 98 etc. Here in
came the problem. If we could trick Gillie to cashing in 100 game,99
games,98 etc., How much money could we win?
That evening I sat with pencil and paper and figured it out
by examining a smaller set of numbers and generalized it to
100(100+1)/2*.05 = $247.5.
Based upon that, this is the summation formula for an
arithmetic progression that I rediscovered when the need rose………