Using the collection of letters blow, you are to construct the word MATHEMATICS by starting at the top of the rhombus and moving down through the letters. At each level, you must move to a letter that is below and to the immediate left or right. Under these restrictions, in how many ways can you construct the word MATHEMATICS?
Here's one possible way to construct the word MATHEMATICS:
The six M's in the middle row can be used as the key to the problem. If you count the number of ways to get to each M in the middle row, you can figure out the total number of paths. There is one path to the M on the left, 5 to the second M from the left, 10 to each of the middle two M's, 5 to the second M from the right, and 1 path to the M on the far right.
Because the rhombus is symmetrical, there are equal numbers of paths from the M's in the middle to the S at the bottom. The one path from the top to the far left M can only be paired with the one path from the far left M to the S at the bottom. Each of the 5 paths from the top to the second M from the left can be paired with the 5 paths from that M to the bottom S. So, there are 
paths from the top to the bottom through the second M from the left. In the same way, there are 
paths through the third M from the left. So, the number of paths from top to bottom through each M is
Adding these values, 
, we get the total number of paths, which is 252.
Solution 2 by Caroline Bjerregaard , Forest Hills Northern High School
To move from the top M to the bottom S, you must make 10 decisions consisting of 5 moves to the right and 5 moves to the left. Therefore, the total number of ways to construct the word MATHEMATICS is 
.
Also solved by: Louisa Avellar, T. C. Heydon, Hudson Thrift (Alamo Heights H.S., San Antonio, TX); Chan-Ik Jang, Matthew Nicholas (Catholic Central H.S.); Jerrod Kempf (Charlotte H.S.); Paige Pearcy (Forest Hills Central H.S.); Erin Sellman (Forest Hills Northern H.S.); Erik Bates, Clare Boals, Brooke Boer, Bridgit Crawford, Mischa Dylewski, Blaire Edgerton, Joan Gary, Eric Hart, Kristin Kopec, Adam Leenheer, Nick Oostendorp, Matthew Parlmer, Kayla Roth, Morgan Smith, Joe Sugiyama, Rebecca Vogelsang, Marc Walker, Sheila Waslawski, Christopher Yang, Alex Zheutlin (Forest Hills Eastern H.S.); Alan Duanmu (Huron H.S.); Robert Harrison (Lansing Catholic H.S.); Spencer Frost, Rachael Juengel, Nate Weller (Linden H.S.); Tim Blaxton, Matthew Ericksen, Kay Haas, Josh Hall, Ashley Meyer, Ian Rhynard, David Schultz (Shepherd H.S.); Preethi Gogineni, Niharika Goparaju, Lucas Tondera (Troy H.S.); Erica Krantz (Williamston H.S.);