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.”
In the set of numbers from 101 to 2,000,000, there were 118,636 “carets” and 87,429 “vees” for a total of 206,065 “carets or vees.” While technically correct with respect to the question as written, a solution stating “there are zero numbers that are carets and vees, as carets and vees are distinct groups” was not chosen as the random winner. The winning solution chosen that of Tim Hurst — congratulations, Tim, you are soon to be the proud owner of a $50 ThinkGeek gift certificate! The rest of us can use the code GEEKDAD59MJ to get $10 off a $50 order at ThinkGeek. Many thanks to ThinkGeek, and to everyone reading and participating in the Puzzle of the Week feature.