Math Chat: The Family Photo and Counting on One Hand

Old challenge (thanks to Norman Goodwin)

A photographer wants to line up the 11 members of the Goodwin family from shortest to tallest (left to right), starting with anyone, adding someone else on either side, and continuing by always adding someone next to those already assembled. How many different ways are there of doing this?


Eric Brahinksy figures that there is 1 way starting with the shortest, plus 10 ways starting with the second shortest (according to which of the next 10 is the short one), plus 45 ways starting with the third shortest (this is already getting hard), plus 120 ways starting with the fourth shortest (this is getting impossible), etc., finally yielding 1+10+45+120+210+ 252+210+120+45+10+1 = 1024 = 210, and in general for a family of n+1, 2n ways.

Norman Goodwin explains that an easier solution is to run the whole process backwards, so you do not have to worry about whom you started with. At each step, you remove someone from either end (two choices), until there is just one person left (one choice). So the total number of possibilities is 2x2x2x...x2x1 = 210.

Question on counting on one hand

Lachlan MacDonald writes and asks, "When people say, 'You could count them on one hand' or 'two hands,' what is the maximum count, culture to culture? Five, 10, 20? I believe it is 19 per hand, using some old illustrations that include the joints and fingertips. Thumbs show two visible joints, fingers three each."

Answer (Victor Hill)

You can count to 9999 on your hands! Howard Eves's "Introduction to the History of Mathematics" (sixth edition, pp. 14, 29) reports that "In time, finger numbers were extended to include the largest numbers occurring in commercial transactions; by the Middle Ages, they had become international. In the ultimate development, the numbers 1, 2, ..., 9 and 10, 20, ..., 90 were represented on the left hand and the numbers 100, 200, ..., 900 and 1,000, 2,000, ..., 9,000 on the right hand."

In his 10th satire, Roman poet Juvenal (AD 60-140) includes the line, "Happy is he indeed who has postponed the hour of his death so long and finally numbers his years upon his right hand," in other words, has reached the age of 100.

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