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# Playing It Smart: The elegant side of roulette

28 October 2008

Roulette has an underlying elegance few solid citizens (or casino bosses, for that matter) appreciate. Bets on combinations such as corners or columns pay the same as on the total spread uniformly over the identical numbers. It's invariably in whole dollars, and it means that the edge is consistent regardless of what's bet.

This holds universally at single-zero roulette. Double-zero games have two possible exceptions. First, the "five-number" line (0, 00, 1, 2, and 3) pays 5-to-1; these spots separately pay 6.2-to-1. Second, even-money "outside" bets return half in some casinos on 0 or 00; money on the coincident 18 spots is a total loss.

To see what this feature means, pretend you drop \$20 on the boundary between two numbers a split. If either number hits, you get 17-to-1 and earn \$20 x 17 or \$340. What if you had \$10 on each number separately? A hit on one pays 35-to-1, grossing you \$10 x 35 or \$350; however, you lose \$10 on the other so your net is \$350 - \$10 or \$340. Just as with the \$20 on the split.

Here's another example. Make believe you put \$1 on each of four three-number streets for instance the rows 1-2-3, 4-5-6, 7-8-9, and 10-11-12. Streets pay 11-to-1. A hit therefore earns you \$11 minus \$1 each on the three misses for a net of \$11 - \$3 or \$8. You could cover the matching 12 numbers for \$4 by betting the "1st 12." This pays 2-to-1, earning you \$4 x 2 or \$8, as before.

This phenomenon follows from the amount paid on a single number and the layout of the betting area. Picture a prototype game with some number of equally-likely outcomes. Say that \$1 on one spot pays \$X and returns the bet. For combinations to pay the same as the separate parts, \$1 split between any two numbers would pay \$(X-1)/2, \$1 on three numbers would pay \$(X-2)/3, and so on.

Check it out for roulette. A \$1 bet on a single number pays \$35. The roulette combinations involve two, three, four, six, 12, and 18 numbers. Some third grade arithmetic and a calculator may help here. The formula gives \$(35-1)/2 = \$17 for two, \$(35-2)/3 = \$11 for three, \$(35-3)/4 = \$8 for four, \$(35-5)/6 = \$5 for six, \$(35 - 11)/12 = \$2 for 12 and \$(35 - 17)/18 = \$1 for 18.

For fun, try the formula for other single-number payoffs, based on the standard combinations and requiring payouts to be in whole dollars. The options are limited. As examples, at 36-to-1, no combinations fit the criteria; 37-to-1 would pay a split 18-to-1 and a 19-number combination 1-to-1, although the latter wouldn't be feasible on the layout; going the other way, 34-to-1 would pay 6-to-1 and 4-to-1 on five-and seven-number combinations, respectively, but neither of these works with a 3 x 12 layout.

At 47-to-1, two, three, four, six, eight, 12, 16, and 24 numbers give whole-dollar payoffs. This suggests a wheel with 49 or 50 grooves and a layout with a 3 x 16 array plus a 0 or a 0 and 00, giving the house 2.04 and 4.00 percent edge, respectively.

Another choice might be 29-to-1. This pays whole dollars for combinations of two, three, five, six, 10, and 15 numbers. A 31- or 32-groove wheel would do it, edgewise. But a problem arises in configuring a layout accommodating enough of these bets to keep the game interesting. Two-number splits are always possible but 4-number corners would be disallowed. The rest might use a 0 or a 0 and a 00 with either three 10-row columns and groups for three, six, and 15, or five 6-row columns for five, 10, and 15 numbers.

An 11-to-1 single-number payoff would return 5-to-1 for two, 3-to-1 for three, 2-to-1 for four, and 1-to-1 for six (1-to-1) numbers. This suggests a 13-position wheel and a 3 x 4 array from 1 through 12 plus a zero. House edge would 7.69 percent.

Maybe you thought a wizened or street-smart old gambling guru pulled the roulette layout and payoffs out of the air. It wasn't that simple. The game doubtless evolved over a long period until something "clicked" and the configuration stabilized. It reflects what the poet, Sumner A Ingmark, insinuated when he inked:

Ideas good and bad oft have differences subtle,
Distinguishing gems from the dross you should scuttle.

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Best of Alan Krigman
Alan Krigman

Alan Krigman was a weekly syndicated newspaper gaming columnist and Editor & Publisher of Winning Ways, a monthly newsletter for casino aficionados. His columns focused on gambling probability and statistics. He passed away in October, 2013.
Alan Krigman
Alan Krigman was a weekly syndicated newspaper gaming columnist and Editor & Publisher of Winning Ways, a monthly newsletter for casino aficionados. His columns focused on gambling probability and statistics. He passed away in October, 2013.