### An Instance of Online Cooperation - the Rascal Triangle

The latest *College Mathematics Journal* brought an absolutely marvelous news. Three young boys from three different countries who never met managed to discover and solve online a problem and report its solution in a hard copy in a respectable math publication.

I remove my hat to

- Alif Anggoro, Java Bekasi, Indonesia, seventh grade
- Eddy Liu, Seattle, USA, eighth grade
- Angus Tulloch, Rimbey, Alberta, eighth grade

The problem arose from a question at an I.Q. Test. Complete the next row in the number triangle

Of course the answer was supposed to be

1 4 6 4 1

Who would hesitate to show off the knowledge of such a beautiful and nowadays popular piece of mathematics? The three boys did! They came up with a different answer and managed to justify it.

As a matter of fact, this kind of "What comes next?" do not have a definite answer. However, there is always a tacit presumption of specific knowledge from which the answer should be drawn. The solver is expected to recognize this background knowledge as a first step in solving the problem. This background provides a natural framework fot the statement and the solution of the problem. Well, said the boys, it's more complicated than that.

Imagine four neighboring elements in the number triangle that form a diamond:

Pascal's triangle is built on the formula South = West + East. The boys came up with a different one:

South = (West × East + 1) / North,

which, produced the following row as an answer to the I.Q. test question

1 4 5 4 1.

The boys observed that, surprisingly, all the numbers came out to be integers, although there was division in the formula. They went on to investigate. The next two rows came out to be

1 5 7 7 5 1

1 6 9 10 9 6 1

Integers again! There was definitely a reason to dig deeper. Look at the triangle

The boys noticed that the NE-SW diagonals exhibit a simple pattern: the m^{th} diagonal (starting the count with m = 0) is an arithmetic progression with m as a difference and 1 as the first term. The entry #n in the diagonal #m is simply *nm + 1*. The prescription to generate the triangle is then

(m + 1)(n + 1) + 1 = [((m + 1)n + 1)(m(n + 1) + 1) + 1] / (mn + 1),

which the boys verified. Do, too.

They boys apparently did not know but the wonderful property of the "diamond algorithm" to produce integer sequences has been first observed by John Conway in the early 1970s and reported by Conway & Guy and Coxeter & Rigby. The more mature mathematicians were interested in the ability of the algorithm to generate integer frieze patterns. The younger mathematicians deserve full credit for observing that the pattern could be extended all the way downwards. Playfully they called their discovery "The Rascal Triangle."

I have in the past colored Fibonacci numbers, the Pascal Triangle, etc in various ways to show visually their prime factorization, whether they are perfect powers, etc.

December 24th, 2010 at 1:32 pmI have now done this for the Rascal Triangle and would like to share it with the 3 boys. I have no way to contact them. perhaps you do.

I could send you some images if you contact me.

Debbie

I know that one of the parents was Andy Liu from University of Alberta. You may try contacting him at

http://www.math.ualberta.ca/Liu_A.html

December 24th, 2010 at 2:55 pm