Fixate on the top figure for 30 seconds and then look at the bottom figure. Though the two circles in the lower figure contain the same number of dots, those on the left appear more numerous. This suggests that the visual system has adapted to the number of items seen in the priming phase, and that, like color, number is a primary attribute of vision.

Psychologists David Burr and John Ross write, “We propose that just as we have a direct visual sense of the reddishness of half a dozen ripe cherries, so we do of their sixishness. In other words there are distinct qualia for numerosity, as there are for color, brightness, and contrast.

“One of the more fascinating aspects of this study … is that although the total apparent number of dots is greatly reduced after adaptation, no particular dots seem to be missing. This reinforces old and more recent evidence suggesting that the perceived richness of our perceptual world is very much an illusion.”

(David Burr and John Ross, “A Visual Sense of Number,” Current Biology 18:6 [March 25, 2008], 425-428.)

From Lee Sallows:

(Thanks, Lee!)

12157692622039623539 = 1¹ + 2² + 1³ + 5⁴ + 7⁵ + 6⁶ + 9⁷ + 2⁸ + 6⁹ + 2¹⁰ + 2¹¹ + 0¹² + 3¹³ + 9¹⁴ + 6¹⁵ + 2¹⁶ + 3¹⁷ + 5¹⁸ + 3¹⁹ + 9²⁰

(The largest such number.)

The dashed disk here has radius 1. Suppose we want to cover it entirely with n smaller disks. How small can those disks be?

Pleasingly, no one has yet found a general answer to this question. If we have only a single covering disk, then obviously it will need to be fully as large as the target. But if we’re allowed six discs, they can do the job even if each has a radius of only 0.555905…, as shown here.

Similar configurations work up to n = 10. But if we’re allowed 11 disks then some creative thinking again becomes necessary to find the best solution. No one has yet found a general strategy that reliably finds the minimum successful size.

One might conjecture that there is an interesting fact concerning each of the positive integers. Here is a ‘proof by induction’ that such is the case. Certainly, 1, which is a factor of each positive integer, qualifies, as do 2, the smallest prime; 3, the smallest odd prime; 4, Bieberbach’s number; etc. Suppose the set S of positive integers concerning each of which there is no interesting fact is not vacuous, and let k be the smallest member of S. But this is a most interesting fact concerning k! Hence S has no smallest member and therefore is vacuous. Is the proof valid?

— Edwin F. Beckenbach, “Interesting Integers,” American Mathematical Monthly, April 1945

The coastline of Nova Scotia was once frequented by pirates, and people occasionally dig for buried pirate treasure. On a local radio program a few years ago I heard an interview with someone who had done a study of attempts to find pirate treasure. He claimed that in most of the cases in which treasure was actually found, it was in a place where treasure-hunters had dug before, rather than in a brand new, previously undug, location. Past diggers simply hadn’t dug deep enough. The previous digger had, in fact, often stopped just short of the treasure. If the previous digger had dug a little deeper than he did, he would have found it.

The interviewer asked him what advice he would give to treasure hunters on the basis of this study; and, producing an interesting application of induction, he lamely suggested that diggers should dig a little deeper than they in fact do. Can you see why this advice is impossible to follow?

— Robert M. Martin, There Are Two Errors in the the Title of This Book, 2002

“The telescope makes the world smaller; it is only the microscope that makes it larger.” — G.K. Chesterton