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Word Triangles

Math brain teasers require computations to solve.

 Puzzle ID: #25007 Fun: (2.72) Difficulty: (2.54) Category: Math Submitted By: phrebh

A word triangle is created by taking a word, putting the first letter on the first line, putting two of the second letter on the next line, three of the third letter on the third line, et cetera. Given the word TEASER, the word triangle would look like this:

T
EE
AAA
SSSS
EEEEE
RRRRRR

If you begin at the top and start spelling TEASER by either moving to the letter directly below the letter you are currently on or by moving to the letter that is immediately to the right of the letter below the letter you are currently on, you could use several different paths to spell the word. For instance, if your path takes you to the third 'A' you could either go to the third 'S' or the fourth 'S', but not to the first or second 'S'.

Can you determine how many possible paths there are for this word triangle? How about for word triangles with words of different lengths?

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 ScooterPants Aug 01, 2005 Good one... I thought the answer was 5! or 120 SaturnCat08 Aug 01, 2005 Tricky One But Another Good One By phrebh. SIMARY1901 Aug 02, 2005 I ALSO THOUGHT THE ANSWER 5 GOOD ONE lisha_kc Aug 03, 2005 I learned this in school...I hate school...but you must be a good student...so praise to you for remembering something... phrebh Aug 03, 2005 Thanks to Gizzer and norcekri who both helped me so much to nail this teaser down. thecowancannons Aug 03, 2005 good job MAroxmysoxonice Aug 06, 2005 i used a tree if anybody has any clue what that is. great teser. i really liked it! javaguru Feb 06, 2009 I didn't read carefully enough at first and started solving the non-trivial problem where you're not restricted from ever moving left. I caught my error before going too far down this road and the answer to the teaser was immediately obvious. However, the original problem I started to solve seemed interesting, so I continued. The solution for this alternate problem for 6 letters is 96. For 1 to 10 letters the solutions are: 1, 2, 5, 13, 35, 96, 267, 750, 2123, 6046 There doesn't appear to be a formula to compute the answer for the nth line that doesn't involve either a summation or a generating function. After failing to crack it I looked it up on OEIS (see http://www.research.att.com/~njas/sequences/ A005773).