Walking Max through some larger numbers, he noticed a few patterns: some numbers increase from digit to digit (such as 1346) and some decrease from digit to digit (such as 9731.) We also noticed that some numbers head one way, and then retreat into the opposite direction. These numbers can be said to “bounce,” and do so exactly once: carets and vees.
For your shot at this week’s $50 ThinkGeek gift certificate, send in the count of counting numbers less than 2 million that are carets and vees. Carets are numbers that increase initially and then decrease, increasing and decreasing for at least one digit, and only “bouncing” one time. Repeated digits are acceptable in determining an increasing or decreasing sequence, i.e., 12231 is a caret, as is 1221, but 1233 and 1335 are not. Vees are the mirror image of carets, save that they initially decrease and later increase, with the same ability to utilize repeated digits, and the same restriction on having exactly one “bounce.”
Send your counts of each to GeekDad Puzzle Central to be included in the random drawing of correct solutions. Good luck!