Archive: March 2026

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?

My lip splits fairly regularly.

Cherry blossoms on the University of Washington campus in Seattle. I love how crowds of people come to campus each spring just to see them!

Handmade panorama. Click and click until it is full-size.

I added a new page to my website.

On it are plots of the roots of all the polynomials in certain sets arising from a given polynomial, P, by taking each coefficient of P and multiplying it by 1 or -1 to create a new polynomial. The plots are all roots of all polynomials generated in this was from a given P. The first is where all of the coefficients of P are 1, so the set of polynomials generated are Littlewood polynomials.

In addition to the plots, I generated sound for each one by treating each root as a sound event. Time runs "right to left" (so the real part of the root determines when its sound event occurs) and the pitch (frequency) is determined (linearly) by the imaginary part of the root. The result is a sound whose spectrogram looks like the plot of the roots.

The range of real parts varies, so the duration of the sounds varies. The extreme case is where the coefficients of P are the factorials of the degree; with such huge coefficients, the roots are quite small, and the result is that the sound is quite short.

A gull.