Tag: dice

new dice problem: #29

I added a new problem (#29 of now 83 problems) to my dice problem collection.

If we roll a die and multiply the results as we go, how many rolls on average will we need until the product is a perfect square?

For a six-sided die the expected number of rolls is 8. Curiously, this is the same as for a five-sided die. The general situation is that the expected number of 2^pi(s) where s is the number of sides of the die and pi(x) is the number of primes less than or equal to x.

Since 6 is not prime, the expected number is the same for 5 and 6-sided dice.

Curiously, it does not matter if the die is fair. As long as, for example, all sides are possible, the expectation is 2^pi(s).

One could ask: for what subsets A of the positive integers does the expected number of rolls until the product is in A depend only on s? This is the case when A is the set of perfects squares. What other A can we consider?

another dice problem

I added another problem and solution to my dice collection. It's problem 77.

Here's the problem statement: "Suppose we play a game with a die. We roll once, and this first roll is the score. We may continue to roll and add to the score, but if the roll ever divides the score we start with (e.g., if our score is 15 and we roll a 1, 3, or 5), then we lose everything and end up with nothing. If instead we choose to stop, we win an amount proportional to the score. What strategy will yield the maximum expected value of our final score?"

This is an optimal stopping problem. I've been interested in optimal stopping problems for quite a while, as a result of these dice problems and other things. However, I haven't found a really good introductory text to the general theory of optimal stopping that really appeals to me, so I end up doing things rather "from scratch" whenever I solve such problems. I feel like I might be missing some useful tools, but I'm not sure that is the case for the simple problems I'm solving.

new dice problem collection addition

Not much of an addition, but I added some probabilities related to the game Can't Stop (which I only learned about recently) to the collection (see problem 73). I calculated these with some simple code to run through all 1296 possibilities, very straightforward. The game is very complicated; see, e.g., this paper by James Glenn and Christian Aloi. I'm not sure what problems related to the game I might add, but I wanted to get these probabilities added for my own curiosity and in case anyone else is interested.

update to my dice problem collection

Updated my dice problem collection. The new problem is problem 40: A die is rolled and summed repeatedly until the sum is 100 or more. What is the most likely last roll? What if we roll two dice at time? Three, etc.?

If one die is rolled, then 6 is the most likely last roll, while 7 is the most likely last roll when two dice are rolled. With three dice, the most likely last roll is 12, but for more dice, a very clear pattern emerges: if the number of dice rolled is odd, then the most likely last roll is the most likely roll (greater than the median), while for an even number of dice, it is one more than the most likely roll. (Here we assume a sufficiently large value for "100": for large numbers of dice, we want to extend the threshold so the analysis is smoother.)

Disquiet Junto 0601

For Disquiet Junto 601, I threw a die in my bathtub and recorded the throws with an AT822 stereo microphone (through a Zoom H5) that I bought (used) years ago but had never used (I’m not really much of a microphone person). Then, using Csound, I placed copies of each recording across about 3.5 minutes, with various densities, filtering, playback speeds and amplitudes. The rolls determined for how much of the piece each recording appears: the rolls were 3,5,6,5,6,3, so the 6 rolls appear throughout, the 5’s appear up to 5/6 of the piece and the 3’s cut off at the half-way point.

More info on Disquiet Junto 601: Disquiet Junto 0601