Inverting the 10-Coin Triangle Puzzle and Other Shapes: A New General Solution
Oscar Atwill ’19 can now add Published Mathematical Discovery to his resume of accomplishments! Oscar, an Eagle Hill School 12th grade student, made a brand new mathematical breakthrough by solving the Reverse the Triangle Puzzle.
This September, in Dr. Tony McCaffrey’s Patterns in Numbers class, students tried to emulate how professional mathematicians work by examining numerical phenomena and looking for patterns. Patterns are first expressed in language and then the language is expressed quantitatively in math. Finally, mathematicians try to demonstrate that the pattern is real—that it holds for all possible cases. In short, this process is known as Play-Pattern-Prove.
Using the Play-Pattern-Prove method, Dr. McCaffrey’s students worked on counting up different shapes of dots (e.g., a square, a cube, a triangle of dots) in various ways to produce different equations. Next, the class moved on to solve the classic Reverse the Triangle Puzzle.
The puzzle starts with 10 coins stacked in the shape of an upward triangle and then inverting the triangle to point downward by moving the fewest number of coins possible. The general solution for any number of rows can be calculated by dividing the number of coins by 3 and ignoring any remainder. A triangle of 4 rows has 10 coins, so 10/3 = 3.333 leads to 3 moves. However, this calculation is fairly nonintuitive as to why it works and requires an inelegant maneuver of periodically discarding a remainder.
In the Reverse the Triangle Puzzle, it was not obvious why the traditional formula worked, so Oscar found a new pattern that is more intuitive to understand. “Oscar dove right into the Triangle of Coins problem and helped discover many patterns about the puzzle. Most of them did not pan out as real patterns, but one did that led to a new, more intuitive equation to calculate the answer to a coin triangle of any size,” said Dr. McCaffrey. “Oscar was impressive in his pattern finding and persistence.”
Using the Play-Pattern-Prove process, Oscar, with Dr. McCaffrey’s help, expressed the pattern in a new equation that always calculates the answer directly and never produces a decimal answer. Further, Oscar and Dr. McCaffrey explored inverting other shapes (e.g., rhombus) and found an interesting connection to the formula for triangles.
Finally, Oscar and Dr. McCaffrey presented a paper summarizing their new mathematical discovery to ArXiv,
which is an electronic repository of published scientific papers owned and maintained by Cornell University.