Page:Carroll - Tangled Tale.djvu/102

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Appendix.

§ 2. The Lodgings.

Problem.—"A Square has 20 doors on each side, which contains 21 equal parts. They are numbered all round, beginning at one corner. From which of the four, Nos. 9, 25, 52, 73, is the sum of the distances, to the other three, least?"

Answer.—"From No. 9."

Let A be No. 9, B No. 25, C No. 52, and D No. 73.

Then AB $\scriptstyle =\ {\sqrt {(12^{2}\ +\ 5^{2})}}\ =\ {\sqrt {169}}\ =\ 13$ ;

⁠AC $\scriptstyle =\ 21$ ;

⁠AD $\scriptstyle =\ {\sqrt {(9^{2}\ +\ 8^{2})}}\ =\ {\sqrt {145}}\ =\ 12+$ ; (N.B. i.e. "between 12 and 13.")

⁠BC $\scriptstyle =\ {\sqrt {(16^{2}\ +\ 12^{2})}}\ =\ {\sqrt {400}}\ =\ 20$ ;

⁠BD $\scriptstyle =\ {\sqrt {(3^{2}\ +\ 21^{2})}}\ =\ {\sqrt {450}}\ =\ 21+$ ;

⁠CD $\scriptstyle =\ {\sqrt {(9^{2}\ +\ 13^{2})}}\ =\ {\sqrt {250}}\ =\ 15+$ ;

Hence sum of distances from A is between 46 and 47; from B, between 54 and 55; from C, between 56 and 57; from D, between 48 and 51. (Why not "between 48 and 49"? Make this out for yourselves.) Hence the sum is least for A.