Remembering Martin Gardner

This week we continue to celebrate the life of Martin Gardner (1914-2010), the popular mathematics and science writer born one hundred years ago this coming Tuesday, Oct. 21. (Check out last week’s post here.) Mr. Gardner was famous for his Mathematical Games column in Scientific American, through which he gave a voice to upcoming mathematicians with fresh ideas. Today we’ll feature one of these mathematicians: Scott Kim, a well-known master of ambigrams — words written in a way that preserves their meaning when flipped over.

I recently asked Mr. Kim about his connection to Mr. Gardner and received this reply by email:

In 1976, I was an undergraduate at Stanford studying music and mathematics. I had fallen in with the lively local branch of the Martin Gardner network, including puzzle collector Stan Isaacs (who is now cataloging Gardner’s correspondence), computer scientist and typophile Donald Knuth, and magician-turned-statistics-professor Persi Diaconis. Together they had urged me to write Martin Gardner, and much to my delight he started publishing my ideas in his column.

I yearned to continue studying mathematics. So I set out on a trip around the country to visit potential graduate schools. In my heart I knew it was a lost cause — no math department could live up to the level of excitement and egalitarian clarity I found in Gardner’s writing. But I went anyway. Sandwiched in between M.I.T. and the University of Toronto, I paid my first in-person visit to Gardner at his home in upstate New York. As it turns out, that visit was the most important part of the trip — I did not go to graduate school in math, but I did go on to become a puzzle inventor and recreational mathematician.

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The decades-long correspondence between Scott Kim and Martin Gardner fills five folders.Credit
Gardner met me at Grand Central Station and drove me to his home. Charlotte [Gardner's wife] greeted me warmly when we got to the house. I was surprised to see no sign of his interests in the living room, aside from an Escher print. But climbing up to the top floor I found myself in Gardner’s small but dense lair, surrounded by file cabinets of all his correspondence. Gardner was a legendary correspondent — everyone I know who wrote him got a prompt and enthusiastic reply — and he kept meticulously cataloged copies of all his letters. What a treasure! I eagerly asked permission to spend some time studying his files and spent some happy hours reading correspondence about the fourth dimension.

Gardner’s files included folders for numbers, where he diligently recorded their curious properties. I showed him my newly invented hobby of writing words [the image at the top of this post was one of the pieces shared] so they read upside down or in a mirror, and he pulled from his files a gem: write the first three digits of pi, being sure to close the top of the 4. Then hold 3.14 up to a mirror. What do you see? The word “PIE”! Delightful.

Years later he published in Games magazine the first puzzle I invented, which also concerned the shapes of letters. Here it is:

I have cut a capital letter out of paper and folded it once. If I unfold it, what letter will you see? Hint: it is not the letter L.

I asked Mr. Kim if he had a favorite Martin Gardner puzzle, and he responded with two:

My two favorite Martin Gardner puzzles are the digital problem, because of its self-referential nature, and the Two-Cube Calendar — because it is a curious, surprising puzzle that comes out of a practical situation. The latter reminds me of the problem of designing two nonstandard dice (not identical), which, when rolled together, have the same probability distribution as two regular dice — another beautiful problem I learned about through Gardner’s column. [We saw these Sicherman Dice --— and met their inventor --— here.] These sort of eccentric, semi-practical problems are related to the spark that powers my ambigram work.

Let’s give them a try. Here are the two puzzles roughly as they appeared in Mr. Gardner’s The Colossal Book of Short Puzzles and Problems:

The Two-Cube Calendar

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In Grand Central Terminal in New York I saw on a store counter an unusual desk calendar. The day was indicated simply by arranging two cubes so that their front faces gave the date. The face of each cube bore a single digit, 0 through 9, and one could arrange the cubes so that their front faces indicated any date from 01, 02, 03 … to 31. On the left cube, you could see two faces whose digits were 1 and 2. On the right cube, you could see three faces whose digits were 1, 3 and 4. What were the four digits that could not be seen on the left cube and the three that could not be seen on the right cube? It is a bit trickier than one might expect.

The Digital Problem

In the 10 cells below, write a 10-digit number so that the digit in the first cell indicates the total number of zeroes in the entire number, the digit in the cell marked “1″ indicates the total number of ones in the entire number, and so forth to the last cell, whose digit indicates the total number of nines in the number. (Zero is a digit, of course.) The answer is unique.

We also have a bonus puzzle this week by Donald Knuth, who, along with Scott Kim, was a featured speaker at a Celebration of Mind event held this past weekend at Stanford University. (Many Celebration of Mind events are still being held this week: Check the site for details.) Dr. Knuth, a mathematician and computer scientist who shared a decades-long friendship with Martin Gardner, offered the following puzzle to mark the centennial of Mr. Gardner’s birth:

Dudeney’s Digital Century Puzzle

There are many curious ways to obtain the number 100 by inserting mathematical operators and possibly also parentheses into the sequence 123456789. Can you come up with one or more of these ways?

Dr. Knuth presented the following as one particularly complicated solution (it’s hard to believe it actually works!). There are many that are far simpler.

100 = (.1 – 2 – 34 × .5)/(.6 – .789)

That’s it for this week’s challenges. As always, once you’re able to read comments for this post, use Gary Hewitt’s Enhancer to correctly view formulas and graphics. And send your favorite puzzles to gary.antonick@NYTimes.com.

Following are several of Scott Kim’s remarkable creations honoring Martin Gardner.