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## An "Indirect" Classic

Each of three blindfolded girls--call them Elsie, Lacey, and Tillie--has a tiny hat placed on her head, a black hat or a red hat invisible to the wearer.

Their instructions: when the blindfolds are removed a girl is to raise her hand if she can see at least one red hat...and raise her second hand if she can prove conclusively the color of her own hat.

Suppose that all three hats are red, plus each girl promptly raises one hand.

Your Task: Outline how one girl, say Elsie, can argue (i.e. prove) that her hat is red.

Source: Gaylord Merriman, To Discover Mathematics, 1942, pp. 23-24

Hint: Set up the situation...try it...role play...and think...make suppositions (i.e. suppose Elsie's hat was black...then...or suppose it was red...then....).

Solution Commentary: Commentary by Gaylord Merriman: "Let Elsie reason indirectly. She wishes to prove that her hat is red; she can do it if the contrary assumption that it is black would necessitate some change in the tableau which now shows each girl with one hand up--for even if Elsie's hat were black each girl would have seen at least one red hat and would have raised a hand. Then (says Elsie) one of the others, say Lacey, could have argued that since Elsie's hat is black abd Tillie's hand is up, Tillie must have raised her hand because Lacey's hat is red. Lacey would thereupon have raised her second hand. So would Tillie, after a similar argument. Yet neither Lacey nor Tillie has raised a second hand. Therefore the known situation pictured by the single up-raised hand of each girl is contradicted by the deductive consequences of the hypothesis that Elsie's hat is black. It is therefore red and Elsie wins the prize..." (p. 23-24)

NOTE: This argument makes the big assumption that Lacey and Tille can be logical as well...a big assumption!