Contingencies Online - July/August 2010

in tribute

aS yoU have no DoUBT heaRD, Martin Gardner died on May 22, at the age of 95. Gardner was a
prolific writer who wrote on numerous subjects, but he was best known as the long-time puzzle editor for
Scientific American. That’s where I first encountered his work, and I’m sure that’s true for many of you, as
well. My seventh-grade math teacher introduced me to his column and his books and they opened my eyes
to how wide and varied the world of mathematics really was. as a kid, if you had asked me what I wanted to
be when I grew up, I would have said, “Martin Gardner.” The fact that I’m currently serving as Contingencies
puzzle editor is a testimony to the influence he had on my life.

And so, in memory of one of the all-time great puzzle editors, I would like to present you three of my favorite short puzzles from Martin Gardner:

1.;A certain desk calendar is made of a stand holding two cubes, with a single digit on each face of the cubes. The day is indicated by arranging the cubes such that their front faces give the date. That is, the cubes can be arranged to indicate 01, 02, 03,..., to 31. What are the digits on each cube?

2.;Suppose you have three boxes, one containing white marbles, one containing black marbles, and one containing a mixture of the two. They are labeled “white,” “black,” and “mixed,” but someone has switched the labels so that every box is incorrectly labeled. you can take one marble at a time out of any box without looking inside, and by this method of sampling, determine the correct label for each box. What is the smallest number of drawings needed to do this?

3.;It will not surprise you at all if I tell you that 2 + 2 = 2 × 2. This is the only set of two positive integers for which this is true; that is n + m <> n × m except if n = m = 2. The puzzle is, can you find a set of three positive integers such that their sum equals their product? How about a set of four integers with this property? How about six? There is a unique solution for each.

Previous issue’s Puzzle
sharing the Wealth
This story took place in my previ-
ous career, as a coder in a dot-com
startup back in the mid-to-late-1990s.
The Internet was a simpler place in
those times, consisting mainly of words
rather than pictures. This was before
the now-common practice of digitally
recording video and audio of one’s every
waking moment, uploading it via a mo-
bile device, and watching the same from
all of one’s friends. But it was still an ex-
citing time to be involved in high-tech
enterprises, finding new and innovative
ways to deliver content across this new
medium and trying to figure out ways
to get people with deep pockets to pay
you for developing new mechanisms for
selling dog food online.

The company had a total of 130 people working there, and, to illustrate what a non-hierarchical and progressive organization it was, the CEO awarded everyone exactly one share of stock. Furthermore, there was a lock-down guarantee (due to the failure to raise any capital) that there would never be another share issued—these 130 shares were it, forever.

One more peculiarity of this being a non-hierarchical and progressive organization was that everything, and I mean every possible issue, was settled by voting. And every single person had one vote. We were entitled to our vote regardless of whether or not we also owned a share of the company. At the beginning, of course, we all did have a share each, but that changed shortly. One final quirk: the CEO was the only one who could call for a vote.

One day, the CEO decided that he wanted to acquire as many shares as he could. The other 129 of us had been there long enough to realize that these shares were unlikely to ever be worth anything, and I personally would have given him mine in exchange for a cup of coffee (16-ounce non-fat latte with four shots of espresso—thanks for asking). But the CEO was both clever and greedy, and he realized that he could gain a lot of shares by asking everyone to vote on reallocating them.