Here are the successive solutions for any number of persons from one to eight:—­

1 = 0 2 = 1 3 = 2 4 = 9 5 = 44 6 = 265 7 = 1,854 8 = 14,833

To get these numbers, multiply successively by 2, 3, 4, 5, etc.  When the multiplier is even, add 1; when odd, deduct 1.  Thus, 3 x 1 — 1 = 2; 4 x 2 + 1 = 9; 5 x 9 — 1 = 44; and so on.  Or you can multiply the sum of the number of ways for n — 1 and n — 2 persons by n — 1, and so get the solution for n persons.  Thus, 4(2 + 9) = 44; 5(9 + 44) = 265; and so on.

268.—­THE PEAL OF BELLS.

The bells should be rung as follows:—­

1 2 3 4 2 1 4 3 2 4 1 3 4 2 3 1 4 3 2 1 3 4 1 2 3 1 4 2 1 3 2 4 3 1 2 4 1 3 4 2 1 4 3 2 4 1 2 3 4 2 1 3 2 4 3 1 2 3 4 1 3 2 1 4 2 3 1 4 3 2 4 1 3 4 2 1 4 3 1 2 4 1 3 2 1 4 2 3 1 2 4 3 2 1 3 4

I have constructed peals for five and six bells respectively, and a solution is possible for any number of bells under the conditions previously stated.

269.—­THREE MEN IN A BOAT.

If there were no conditions whatever, except that the men were all to go out together, in threes, they could row in an immense number of different ways.  If the reader wishes to know how many, the number is 455^7.  And with the condition that no two may ever be together more than once, there are no fewer than 15,567,552,000 different solutions—­that is, different ways of arranging the men.  With one solution before him, the reader will realize why this must be, for although, as an example, A must go out once with B and once with C, it does not necessarily follow that he must go out with C on the same occasion that he goes with B. He might take any other letter with him on that occasion, though the fact of his taking other than B would have its effect on the arrangement of the other triplets.

Of course only a certain number of all these arrangements are available when we have that other condition of using the smallest possible number of boats.  As a matter of fact we need employ only ten different boats.  Here is one the arrangements:—­

1 2 3 4 5
1st Day (ABC) (DBF) (GHI) (JKL) (MNO)
8 6 7 9 10
2nd Day (ADG) (BKN) (COL) (JEI) (MHF)
3 5 4 1 2
3rd Day (AJM) (BEH) (CFI) (DKO) (GNL)
7 6 8 9 1
4th Day (AEK) (CGM) (BOI) (DHL) (JNF)
4 5 3 10 2
5th Day (AHN) (CDJ) (BFL) (GEO) (MKI)
6 7 8 10 1
6th Day (AFO) (BGJ) (CKH) (DNI) (MEL)
5 4 3 9 2
7th Day (AIL) (BDM) (CEN) (GKF) (JHO)

It will be found that no two men ever go out twice together, and that no man ever goes out twice in the same boat.