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.)