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Euler's analysis continued with a discussion of the fairness of the bank's increasing profit margin. It is important to note here a significant difference between the Genoese lottery and most modern versions of the lottery. The Genoese lottery (and, for that matter the contemporary Italian Lotto) offers fixed-odds payoffs. That is, the player knows in advance how large a prize is at stake, irrespective of the number or distribution of tickets sold. In the long run, the bank is guaranteed to win, and to win big. However, if the state were to have a run of bad luck, and a relatively large number of major prizes were to be awarded in a drawing where relatively few tickets had been sold, the bank could, in principle, be broken.

The danger of bankruptcy is avoided in lottery games played today in the USA and most other countries by adopting a version of the method of pari-mutuel betting used in horse racing. The term, literally meaning `mutual wager', comes from two French words. A portion of each wager placed is set aside to cover the cost of running the lottery and the state's revenue, and the remainder is placed in a pool, to be paid out to the winners. In effect, players are betting against each other, with the state taking a cut of every wager.

In New York State's Lotto, for example, 6 winning numbers are chosen from a wheel containing 59 numbered balls. Players choose 6 numbers between 1 and 59, and win a cash prize if 3, 4, 5 or 6 of their chosen numbers match the winning numbers drawn from the wheel. (As an added wrinkle, a seventh "bonus number" is also drawn from the wheel, and a special second prize goes to any player who matches 5 of the 6 regular numbers drawn, plus the bonus number.) Thirty-eight cents out of every one-dollar ticket goes into a pari-mutuel pool, and it is from this pool that all prizes are paid. Three-quarters of that amount is reserved for the jackpot, paid to players who match all 6 numbers of the winning numbers. If there is no jackpot winner in a particular drawing, the jackpot rolls over, making the subsequent drawing even more attractive to players. Reliable estimates on the size of the jackpot are available before the drawing, but only at the close of betting is the exact payoff known.

In Euler's time, the technology needed to deliver the sort of up-to-the-minute information on ticket sales used in the calculation of pari-mutuel odds was not available. Therefore Euler noted in his letter to Frederick that it was entirely proper for the bank to offer relatively smaller payoffs for the riskier *ambo* and *terno* bets in order to insulate itself from calamity. He also observes that if the total number of bets is small, it would be undesirable to have many players choosing the same *ambo* or *terno* bets, although it's not clear what sort of practical bookkeeping procedures might have been instituted at that time to avoid these multiple bets.

We note that although Roccolini's proposal did not allow for a gambler to bet on all 5 numbers in the drawing, this sort of bet was permitted in other Genoese-style lotteries of the 18th century, including the lottery that was eventually instituted in Berlin in 1763. In fact, it was even possible in the French Royal lottery to bet on all five numbers **and** the order in which they were drawn. The payoff, at a million to one, made this game irresistibly attractive to the French populace, particularly the lower classes, despite the fact that the true odds are in excess of five billion to one. "There were occasions when mathematicians wrote learned articles demonstrating why people should not play the lottery at those odds," writes Katz [9, p. 598] in describing the French lottery, "but the only mathematicians to whom people paid attention were those who sold sure-fire methods for picking the winning numbers!"

Euler neither wrote cautionary epistles, nor did he hawk sure-fire winning strategies. Instead, the Genoese lottery inspired in him a series of mathematical articles concerning some of the trickier questions of combinatorics and probability theory to be tackled in the 18th century. The first glimmer of these deeper thoughts is his observation, towards the end of the letter of September 17, that there is nothing sacred about the numbers 5 and 90; the Genoese lottery may be abstracted to any number *n* of numbered balls or tickets in the urn, and any number *t* < *n* of such tokens chosen in the drawing. He closes by outlining for Frederick an alternate lottery scheme, involving *n*=100, *t*=10, where players bet on 1, 2, 3 or 4 numbers.

Robert E. Bradley, "Euler's Analysis of the Genoese Lottery - The Royal Charge (continued)," *Convergence* (August 2010)