Marilyn needs too many Ping-Pong Balls

Marilyn is Wrong Copyright © 1997-1998 Herb Weiner. All rights reserved.

Ask Marilyn ® by Marilyn vos Savant is a column in Parade Magazine, published by PARADE, 711 Third Avenue, New York, NY 10017, USA. According to Parade, Marilyn vos Savant is listed in the "Guinness Book of World Records Hall of Fame" for "Highest IQ."

In her Parade Magazine column of January 5, 1997, Marilyn claimed that 4900 balls would be required to build a four sided equilateral pyramid (with a square base and equilateral triangles on the four sides), despite the fact that five balls would be quite sufficient. In her column of March 2, 1997, Marilyn attempted to justify her original answer by claiming that the balls had to be arranged in a square before building the pyramid.

Let's try to economize, please and Don Groves <> both wrote to suggest that five balls are sufficient: a square base consisting of four balls, with one ball on top. Each of the four sides would be an equilateral triangle consisting of three balls.

An Interesting Theory

Fred Moolten <> wrote to suggest an interesting theory:
How does Marilyn arrive at answers to readers' questions? PARADE might wish readers to think that she uses her immense store of knowledge to answer factual questions and her enormous reasoning power to solve puzzles. Often, however, a clue to how a process works is provided when the process misfires. Take one of her latest gaffes -- the ping pong ball puzzle. A reader asked her to state the smallest number of ping pong balls needed to build a pyramid by adding successive layers to a square base. As some of your respondents have noted, the correct answer is 5, which can be thought of as one squared + two squared. The next higher pyramid requires 14 (one squared + two squared + three squared), then 30, and so on, with Marilyn's answer, 4900, way off in the distance. All of these numbers are termed "pyramidal numbers" for obvious reasons. Could she have reasoned her way to 4900, overlooking all the intervening pyramidal numbers along the way? I can't imagine how.

On the other hand, what happens if instead of asking her brain what the answer is, she asks a database (or asks her secretary to ask a database)? Three critical elements of the reader's question were "number", "pyramid", and "square". Ask a database (e.g. AltaVista, using the Advanced search) to search for "pyramid AND number AND square" and you find that one of the early links to come up is Great Pyramid Courses. (Click in the Advanced box at the top of the Alta Vista Search page, enter the Selection Criteria "pyramid AND number AND square", then enter the Results Ranking Criteria "pyramid" and click "Submit Advanced Query". The above link currently comes up near the top of the second page of results, but this could change in the future as the number of web pages grows.) Checking out this link, you find that there is only a single number that is both pyramidal and also a square. What number? You guessed it. Does this give us some hint of how Marilyn solves things? Perhaps that's a question a reader should ask her for the next issue of PARADE.

You can quickly find even more items on this by typing "pyramidal numbers", but that's cheating, since the PARADE puzzle didn't use that term

An alternative solution

Mark Lyttle <> wrote to point out that any pyramidal number that is an integral multiple of four will also work, if you arrange them on the ground so that they form a hollow rather than a solid square. The smallest such number, 140 (1 + 4 + 9 + 16 + 25 + 36 + 49), can be arranged using four sets of 35 balls each to form a hollow square made up of 36 balls on each side (the corner balls are part of two adjacent sides). Then, these same balls can be used to create a pyramid with a 7 x 7 base, and sides made up of triangles with seven balls on each side. There are many other numbers that will also work. last updated September 20, 1998 by