Page:Carroll - Tangled Tale.djvu/118

 

ANSWERS TO KNOT V.

Problem.—To mark pictures, giving 3 ×'s to 2 or 3, 2 to 4 or 5, and 1 to 9 or 10; also giving 3 o's to 1 or 2, 2 to 3 or 4 and 1 to 8 or 9; so as to mark the smallest possible number of pictures, and to give them the largest possible number of marks.

Answer.—10 pictures; 29 marks; arranged thus:—

×	×	×	×	×	×	×	×	×	∘
×	×	×	×	×		∘	∘	∘	∘
×	×	∘	∘	∘	∘	∘	∘	∘	∘

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Solution.—By giving all the ×'s possible, putting into brackets the optional ones, we get 10 pictures marked thus:—

×	×	×	×	×	×	×	×	×	(×)
×	×	×	×	(×)
×	×	(×)

By then assigning o's in the same way, beginning at the other end, we get 9 pictures marked thus:—

							(∘)	∘
					(∘)	∘	∘	∘
(∘)	∘	∘	∘	∘	∘	∘	∘	∘

All we have now to do is to run these two wedges