A roulette player had a system of playing one dollar 7 times, red or black, then 7 dollars 7 times, red or black, then 49 dollars 7 times, red or black, etc., each time 7 bets in increasing powers of 7.
How many times had he won if he finally won net 777,777 dollars?
For those not familiar with the roulette, in this particular bet, if you bet 1 dollar, either you lose it or gain another 1 dollar.
The powers of 7 make the total of the sets of trials act like a base7 number, except that the possible values at each base7 position are 7, 5, 3, 1, 1, 3, 5, 7, rather than the usual 0 thru 7.
Two things must be done to change this: calculate how much the total loss would be if the player had lost every roll of the ball and add that to the total winnings to see how much he had to make up, and then divide by two, as he gains 2 position values for each win.
First we must figure how many base7 positions are needed. 777777 is 7^6.9706..., so we'll assume a 7digit base7 number. (This is an odd number, as it needs to be, as, if an even number of rounds occurs, in which each gain/loss is odd, then the winnings would be even.) If the player had lost $7 in the first set of 7 plays and $49 in the next set, etc., then he would have lost $960,799 altogether in 7 sets of 7 plays. But he didn't lose this; in fact, he gained 777,777, so he must have overcome this by 960799 + 777777 = 1738576.
As mentioned, regardless of the stakes involved in each set of plays, each win also removes a loss, so this amount needs to be halved before converting to base7. That gives us 865288. In base7 that's 7250240, and as we had surmised, it's a 7digit base7 number.
The zero at the righthand end represents having won zero trials in the first set of 7 spins. The 4 to its left represents four wins in the second round at $7 per roll/spin, etc.
In all, then, he had 7+2+5+0+2+4+0 = 20 wins.
Let's check if this works out:
0 wins 7 losses at $1 7
4 wins 3 losses at $7 +7
2 wins 5 losses at $49 147
0 wins 7 losses at $343 2401
5 wins 2 losses at $2401 +7203
2 wins 5 losses at $16807 50421
7 wins 0 losses at $117649 +823543

net winnings 777777

Posted by Charlie
on 20080713 18:31:33 