Scott Blomquist

Two really cool dice puzzles that have already been solved to death by bored mathematicians:

Nontransitive dice

Number three blank dice such that one of the dice (say die A) beats another of the dice (die B) more than half of the time, which in turn beats die C more than half of the time. Nothing unusual so far, right? Now make it so that die C beats die A more than half the time as well. Yes, put in simple terms, A > B > C > A. Mathematicians call these nontransitive dice.

Imagine the great bar bets that you can make with these: "choose any of these three dice; then I’ll choose one; we’ll roll 10 times, and whoever rolls a higher number more times wins $20–all you have to do is figure out which die is best."

For extra fun, design a set of dice that enable you to offer to take on two guys at once who each choose a die from a pool of 7 dice, and then you choose a remaining one that beats them both!

Try to come up with at least the 3 dice set on your own, and then check out the Tournament Dice article on Math Games for lots of fun information on both of the above schemes.

"Normal" dice with abnormal numbers

Next, number two 6-sided dice in such a way that if you roll them both and take the sum, it’s the same distribution of sums as that of two normal 6-sided dice (i.e. 1 way to roll a 2 or 12, 2 ways to roll a 3 or 11, and so on). It’s cheating to use non-positive numbers (otherwise, you could just add N to each face on one die and subtract the same N from each face on the other).

I found this one easier to contruct on my own than the nontransitive dice. Give it a shot and then see Wolfram’s Sicherman Dice write-up for more details on this.

Sergey once asked a large assemblage of Googlers what our greatest corporate expense was. “Health insurance!” was one answer shouted back. “Salaries!” “Servers!” “Taxes!” “Electricity!” “Charlie’s grocery bills!,” came back others. “No,” said Sergey. “Opportunity cost.” He explained that the products we weren’t launching and the deals we weren’t doing threatened our economic stability more than any single line item in the budget. It became a regular call and response at staff meetings and added to the sense that no matter how hard we were working, success was slipping through our fingers. Rather than cause employees to feel defeated, however, it became a rallying cry to redouble their efforts.

In other words, "The opportunity cost of our employees sleeping and having lives outside work is our company’s greatest expense." How could that possibly encourage super-human commitment rather than induce demoralizing levels of despair?! Is this really a strong motivator if the prospect of having-a-cooler-project-at-work-than-any-spare-time-project-could-possibly-be is not? (I’m willing to consider the possibility that my mistake here is to assume that the average Google employee has a job as cool as mine in MSN Search.)

Arthur, I’m especially interested in your thoughts on this.

[Edit: fixed my misspelling of Sergey's name.]

I don’t vote, I already have a religion, and I hate whales.