Here are three problems, all original with me, that you can solve in your head. They're perfect for keeping from being bored when you're out walking or otherwise unable to safely read or do other tasks. Inspired by Sloane's strange-looking A258107 (2, 3, 4, 82000), I thought about the smallest number (other than "1") that's represented by nothing but odd digits in the first N bases (binary, ternary, etc.). I quickly proved to myself that no number has this property for bases 2, 3, and 4. What about just in odd bases? I quickly proved to myself that no number (except "1") has this property in bases 3, 5, 7, and 9. I had been saving the dead president's problem for just after the next president died. But I've decided that would be in bad taste. So here goes: There are currently 38 dead US presidents. Their dates of birth and death are all known. If I sort them in order of birth, they're in a certain order, starting with Washington and ending with Kennedy. If I sort them in order of death, they're in a slightly different order, starting with Washington and ending with Ford. Suppose I sort them by the date on which they had been (or will have been) dead longer than they were alive? I get yet a third slightly different order. (Once again it begins with Washington and ends with Ford. So does sorting them by when they will have been dead twice as long as they were alive.) The question is: How many different sort orders are there, as you vary the parameter X for "dead X times longer than alive" over all real numbers, including negatives. Yes, you can solve it in your head, and no, the answer isn't 38 factorial. All of these puzzles of course give rise to more challenging puzzles.