You are at The Riddler Casino, and you are betting on a horse race. The casino provides betting odds (in the American format) for each horse. For example, odds of -150 means that for every \$150 you bet, you win an additional \$100. Meanwhile, odds of +150 means that for every \$100 you bet, you win an additional \$150.
To break even, a horse with -150 odds should win 60 percent of the time, while a horse with +150 odds should win 40 percent of the time. (Yes, both +100 and -100 correspond to a 50 percent chance of victory.) Of course, most casinos rig the odds such that betting on all the horses in a race will cause you to lose money.
But not The Riddler Casino! Here, a horse with -150 odds has exactly a 60 percent chance of winning, and a horse with +150 odds has exactly a 40 percent chance.
Today, a five-horse race has caught your eye. The odds for three of the horses are +100, +300 and +400. You can’t quite make out the odds for the last two horses, but you can see that they’re both positive multiples of a hundred. What are the highest possible odds one of those last two horses can have?
</div>The corresponding win for odds of +300 is 25% and for odds of +400 is 20%.
The remaining two horses, with odds that are multiples of 100, have a corresponding percent sum = 5%.
$$\dfrac{100}{100a+100} + \dfrac{100}{100b+100} = 0.05$$
$$\dfrac{1}{a+1} + \dfrac{1}{b+1} = \dfrac{1}{20}$$
$$20(b+1) + 20(a+1) = (a+1)(b+1)$$
$$400 = (a+1)(b+1) - 20(a+1) - 20(b+1) + 400$$
$$400 = (a+1-20)(b+1-20) = (a-19)(b-19)$$
The highest possible odds occurs when, WLOG, $a-19 = 1$ and $b-19 = 400$.
Therefore $a = 20$ and $b = 419$.
The five horses are thus:
| Odds | Percent |
|---|---|
| +100 | 50% |
| +300 | 25% |
| +400 | 20% |
| +2000 | ≈ 4.76% |
| +41900 | ≈ 0.238% |