The drawing below, brazenly swiped from a 1964 Martin Gardner column, illustrates the solution to a well-known puzzle. If you stack n bricks on a table, how far can you make them extend over the edge without toppling?

The answer given, for bricks of unit length, is one-half the nth harmonic number, the sum of the […]

Among the 250 million Rubik’s cubes manufactured since 1980, how many lie abandoned in a scrambled state, having never regained their original configuration since being taken out of the box? Most of them, I would guess. Now comes word that those cubes might be restored to pristinity with a little less effort. The upper bound […]

The March-April issue of American Scientist is now available on the Web; paper copies should be on their way soon. My column is about hump yards and turnouts and wyes—in other words, about algorithms for railroad workers. “Computing with locomotives and box cars takes a one-track mind.” There’s a small puzzle near the end of […]

Our story so far: Having stumbled upon the Jacobsthal numbers, 1, 3, 5, 11, 21, 43, 85, 171, 341,…, I idly asked, “Who was Jacobsthal?” Keith Matthews promptly responded with a wealth of biographical information, even arranging to have an obituary translated from the Norwegian. So I asked, “Where did Jacobsthal mention these numbers?” and […]

In an item published here last May I stumbled across the sequence

1 3 5 11 21 43 85 171 341 683 1365 2731 5461 10923 21845 43691
87381 174763 349525 699051

which I dubbed “the oddest numbers” but which Neil J. A. Sloane’s Encyclopedia of Integer Sequences calls the Jacobsthal sequence. I asked: Who is or […]

Three postings here (1, 2, 3) have discussed what happens when you form all pairwise sums and differences from a finite set of integers. The number of differences almost always exceeds the number of sums—a fact that lends special interest to the occasional exceptional sets, with More Sums Than Differences (MSTD).
A question left up in […]

I’ve received the following note from Barry Cipra, bit-player’s Bureau Chief in Northfield, Minnesota, (where all hail broke loose yesterday):

Your latest postings [here and here] have motivated me to idle away some time with a variant of the problem(s) you discussed. To wit, I got to wondering what happens if you restrict yourself to taking […]

Kevin O’Bryant, whose work on sets that have more sums than differences was mentioned in this recent post, writes as follows:

Here’s a related problem that Mel Nathanson and myself (with Ruzsa and a few students) have also been thinking about. For a finite set A of integers, let S = {a + b : a,b […]

Take a set of integers, say {0, 2, 5, 8, 11}, and write down all the numbers that can be represented as sums of two elements drawn from this set. For our example the answer is {0, 2, 4, 5, 7, 8, 10, 11, 13, 16, 19, 22}. Now construct the corresponding set of pairwise […]

At the library the other day I was perusing the Collected Rantings and Ravings of Edsger W. Dijkstra [Note 1]. While leafing through the pages in search of something else [Note 2], I stumbled across “An Exercise for Dr. R. M. Burstall” [Note 3], a brief essay in the form of an open letter, written […]