I’ve written quite a few posts on mental math over the years. I think mental math is important/interesting for a couple reasons. First, there is some utility in being able to carry out small calculations with rough accuracy without having stop and open up a calculator. Second, the constraints imposed by mental calculation make you look at a problem differently, often in interesting ways. An approximate mental calculation often requires more understanding than a precise machine calculation.
Divisibility and factoring
- Divisibility by 2, 3, 4, …, 13
- Divisibility by base minus 1
- Divisibility by base plus 1
- John Conway’s factoring tricks
- Divisibility by any prime
- Fermat’s factoring trick
Day of the week
Transcendental functions
This page outlines how to calculate logarithms base 2, e, and 10, and their inverses, linking to other posts for details.
It also outlines computing trig functions, roots, and the gamma function. See the links below for more accurate ways to compute logarithms.
Incidentally, when I’ve posted about these approximations before, inevitably someone will say these are just Taylor approximations. No, they aren’t.
Speaking of mental math, I just read your post on tetrahedral numbers https://www.johndcook.com/blog/2025/05/30/stacking-cannonballs/
Without pencil or paper, I realized that if you allow negative tetrahedral numbers, 17 can be written as the sum of 4 with a much simpler se5t of numbers than the one you posted: 17 = T(4,3)+T(-3,3)+T(-3,3)+T(-3,3) (20, -1, -1, -1)
I told my daughter about negative tetrahedral numbers, and she almost immediately found another combination of 4 that add up to 17, then a combination of 3 that does, too:
17 = T(3,3)+T(2,3)+T(2,3)+T(-3,3) (10, 4, 4, -1)
17 = T(4,3)+T(-4,3)+T(1,3) (20, -4, 1)