Writing in the north of England in the early 8th century, the Venerable Bede described a Roman system of finger counting:

1 = the little finger bent at the middle joint
2 = the ring and little fingers bent at the middle joints
3 = the middle, ring, and little fingers bent at the middle joints
4 = the middle and ring fingers bent at the middle joints
5 = the middle finger only bent at the middle joint
6 = the ring finger bent at the middle joint
7 = the little finger closed on the palm
8 = the ring and little fingers closed on the palm
9 = the middle, ring, and little fingers closed on the palm
10 = the tip of the index finger touching the middle joint of the thumb
11 to 19 = the actions denoting each numeral from 1 to 9 plus that of 10
20 = the thumb tucked between the index and middle fingers, so that the thumbnail touches the middle joint of the index finger
21 to 29 = the actions denoting each numeral from 1 to 9 plus that of 20
30 = the tips of the thumb and index finger touching and forming a circle or ring
40 = the thumb and index finger standing erect and close together
50 = the thumb bent at both joints and held against the palm
60 = the index finger closed over the thumb
70 = the first joint of the index finger resting over the first joint of the thumb, which is held nearly straight
80 = the tip of the index finger resting on the first joint of the thumb
90 = the thumb bent over the first joint of the index finger

The signs for 100, 200, 300, and so on are the same as 10, 20, 30, but made by the right hand; and the signs for 1,000, 2,000, 3,000 and so on are the same as 1, 2, 3 but made by the right hand. “To add two numbers, one simply signed the first, then made the mental arithmetical calculation and reproduced the gesture corresponding with the correct sum,” writes Angus Trumble in The Finger: A Handbook (2010). “The process was cumulative; to add a further number to the sum of the first two, you proceeded to represent the gesture corresponding with the new total, and so on. Likewise, the task of subtraction merely threw the whole system into reverse. It was perfectly clear to anyone observing you carry out these separate procedures whether the job in hand was one of addition or subtraction.”

Trumble says that at the end of the 19th century Wallachian peasants were discovered to have preserved a few methods of digital multiplication and division that had been preserved throughout the Roman empire. Here’s one.

Choose one of these cards and fix it clearly in your mind. Then open the answer box.

Suppose you want to hang a picture by a string that’s attached at two points on the back of the frame. How can you arrange the string on two nails such that the picture will fall if either nail is removed?

One solution is above. I don’t know who first asked the question; I first saw it in Mathematical Mind-Benders, by Peter Winkler, who got it from Giulio Genovese, a mathematical graduate student at Dartmouth, who’d seen it in more than one source in Europe.

But it opens up a surprisingly rich discussion — see the paper below for some entertainingly complex variants.

(Erik D. Demaine et al., “Picture-Hanging Puzzles,” Theory of Computing Systems 54:4 [2014], 531-550.)

54199³ = 159211275242599 and 15921 + 12752 + 42599 = 71272

71272³ = 362040234715648 and 36204 + 02347 + 15648 = 54199

(From Edward Barbeau’s Power Play, 1997.)

Éric Angelini devised this progressively self-inventorying array:

10 71 32 23 14 15 16 27 18 19

20 81 72 53 44 35 26 47 38 29

40 101 82 73 64 65 56 77 58 39

60 131 92 93 74 75 86 107 88 69

80 201 122 113 84 85 96 117 138 89

The first line describes its own contents: It contains one 0, seven 1s, three 2s, and so on.

In the same style, the second line describes the joint contents of lines 1 and 2.

And so on: The fifth line describes the contents of the whole table: It contains eight 0s, twenty 1s, twelve 2s, etc.

He suspects that a six-line table is possible, but he hasn’t found one yet.

More here.

(Thanks, Éric.)

1² + 2² + 3² + … + 24² = 70²

This is the only case in which the sum of the first k perfect squares is itself a square.

03/23/2020 UPDATE: Reader Pieter Post made a pyramid of 4900 golf balls in the Netherlands last summer:

It took him an hour and a half. (Thanks, Pieter.)

In a 1988 experiment with 2-year-olds, psychologist Alan Leslie asked each child to “fill” two toy cups with imaginary “juice” or “tea” from a bottle. Leslie then said, “Watch this!”, upended one of the cups, shook it, and replaced it next to the other cup. Then he asked the child to point to the “full cup” and the “empty cup.” Though both cups had been empty throughout, all 10 of the 10 subjects indicated that the “empty” cup was the one that had been inverted.

“This leads to pretending something that is true, namely, that the empty cup is empty,” Leslie wrote. “At first glance, this may seem ridiculous. But there is, of course, an important difference between the empty cup is empty and pretending (of) the empty cup ‘it is empty.'” Children distinguish between pretense and reality even when the content of those beliefs is the same.

“These examples help us realize that, far from being unusual and esoteric, cases of ‘non-counterfactual pretence’, that is, pretending something is true when it is true, are ubiquitous in young children’s pretence and indeed has an indispensable role in the child’s ability to elaborate pretend scenarios.”

(Alan M. Leslie, “Pretending and Believing: Issues in the Theory of ToMM,” Cognition on Cognition [1995], 193-220.)