Mathematical recreations and essays . e consists merely of two ends hooked together. If this random coupling gives us one single curve then theproposition is proved; for starting at any point we shall goalong every branch and come back to the initial point. But 174 UNICURSAL PROBLEMS [CH. IX if this random coupling produces anywhere an isolated loop, L,then where it touches some other loop, M, say at the node P,unfasten the four hooks there (viz. two of the loop L and twoof the loop M) and re-couple them in any other order: thenthe loop L will become a part of the loop M, In this way,by alteri
Mathematical recreations and essays . e consists merely of two ends hooked together. If this random coupling gives us one single curve then theproposition is proved; for starting at any point we shall goalong every branch and come back to the initial point. But 174 UNICURSAL PROBLEMS [CH. IX if this random coupling produces anywhere an isolated loop, L,then where it touches some other loop, M, say at the node P,unfasten the four hooks there (viz. two of the loop L and twoof the loop M) and re-couple them in any other order: thenthe loop L will become a part of the loop M, In this way,by altering the couplings, we can transform gradually all theseparate loops into parts of only one loop. For example, take the case of three isles, A, B, C, eachconnected with both the others by two bridges. The mostunfavourable way of re-coupling the ends at A, B, G would beto make ABA, AC A, and BOB separate loops. The loopsABA and AC A are separate and touch at A; hence we shouldre-couple the hooks at A so as to combine ABA and AG A into A. one loop ABACA. Similarly, by re-arranging the couplingsof the four hooks at B, we can combine the loop BCB withABACA and thus make only one loop. I infer from Eulers language that he had attempted tosolve the problem of giving a practical rule which wouldenable one to describe such a figure unicursally withoutknowledge of its form, but that in this he was however added that any geometrical figure can be de-scribed completely in a single route provided each part of itis described twice and only twice, for, if we suppose that everybranch is duplicated, there will be no odd nodes and the figureis unicursal. In this case any figure can be described com-pletely without knowing its form: rules to effect this aregiven below. Third. A figure which has two and only two odd nodes comhe described unicursally by a point which starts from one of theodd nodes and finishes at the other odd node. OH. IX] UNICURSAL PROBLEMS * 175 This at once redu
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