1. A word puzzle:
"Downshifting" a word means replacing its initial letter with the immediately preceding letter of the alphabet. For example, to downshift the word BOIL, replace the B with an A to obtain AOIL. (Unfortunately, AOIL is not a word.)
Some words are "downshiftable," meaning that when you downshift them, you obtain a new word. For example, TIP is downshiftable, because it downshifts to SIP.
In fact, SIP is itself downshiftable, because it downshifts to RIP.
But unfortunately, RIP is not downshiftable, because QIP is not a word.
Altogether then, we might say that TIP was downshiftable twice: TIP -> SIP -> RIP.
Can you find a word that is downshiftable three or more times?
The best word I could think of before I fell asleep last night was 3x downshiftable. When I woke up this morning, I wrote a program to see if my computer could do any better. It found six words that were 3x downshiftable (mine was in the list), as well as a single 4x downshiftable word! Happy hunting.
Fine print: In case you were wondering what to do with words that begin with A, let's agree that A always downshifts to Z, as in AERO -> ZERO. (Not that AERO is a legitimate word, it's just the best example I could think of to illustrate the concept.)
2. A word problem. This one is for math-major types (solution here).
John picked up a notebook and saw that there was a positive real number written on each page. The product of all the numbers was e^(S/e), where S was the sum of all the numbers. Prove that S is not an integer, and find the average of the numbers in the notebook.