Second Method. Number one heads the outside column. Solutions only possible when 12 and 14 of the middle numbers are diagonally adjacent. Bring the 11 alongside of the 12.

I. The middle numbers occur as some combination of Table I.

1. The increase of outside numbers with motion of watch hands; a 3,

and c 1.

2. Increase of outside numbers against motion of watch hands; d 1,

and b 2.

II. The middle numbers some combination of Table IT.

1. Increase of outside numbers wiih motion of watch hands; b 3,

and d 1.

2. Increase of outside numbers against motion of watch hands; c 2,

and a 1. Number fifteen leads. Solutions only possible when 1 and 3 are diagonally adjacent.

Bring the 5 alongside the 4.

III. The middle numbers occur as some combination of Table III.

1. Decrease of outside numbers with motion of watch hands ; b 2,

and d 1.

2. Decrease of outside numbers against motion of watch hands; a 3,

and c 3.

IV. Middle numbers occur as some combination of Table IV.

1. Decrease of outside numbers with motion of watch hands; a 3,

and c 1.

2. Decrease of outside numbers against motion of watch hands; b 2,

and d 2.

Third Method. The numbers read from left to right on the first and third lines and from right to left on the second and fourth, or vice versa, thus:

12 3 4

8 7 6 5

9 10 11 12
15 14 13

The order of outside numbers is 1. 2, 3, 4, 8, 7, 6, 5, 9, 10, 11.

Solutions are only possible when 12 and 15 or 1 and 4 are diagonally adjacent. The first four numbers pass 13-12 in I and II, and 3-4 in III and IV, in the order named, the head of the column coming to rest at the square diagonally adjacent to the 12 or the 4 respectively.

Number one heads the column. The upper line is full. I. The middle numbers a combination of Table II.

1. First four numbers increase with motion of watch hands ; a 2, and

d2.

2. First four numbers increase against motion of watch hands; b 1,

and c 3.

II. Middle members a combination of Table I.

1. First four numbers increase with motion of watch hands ; b 3, and

c3.

2. First four numbers increase against motion of watch hands ; a 2,

and d 2. Number fifteen heads the column. Order of outsiders, 15, 14, 13, 12, 8, 9, 10, 11, 7, 6, 5.

III. Middle numbers a combination of Table 1V.

1. Last tour numbers decrease with motion of watch hands ; a 1, and

d3.

2. Last four numbers decrease against motion of watch hands ; b 3,

and c 2.

IV. Middle numbers a combination of Table III.

1. Last four numbers decrease with motion of watch hands ; b 1, and

c2.

2. Last four numbers decrease against motion of watch hands ; d 3,

and a 2.

It is thus seen that there are four tables, each containing twelve combinations of the middle numbers or 48 combinations in all. Each of the three methods of solution takes four combinations from each table or one from every horizontal line, and no combination will permit of but one solution. Since these are all the possible combinations and a solution is given for every one it follows that no other solutions are possible than those above given.

It is but just to say that the first demonstration of the possible solutions of the first method was printed by me in the Bulletin of Feb. 26; showing that in the 13, 15, 14 difficulty position, two solutions were possible, but that the box must be turned if the 1 was to occupy the left hand upper square. Afterwards a paraphrase of this was printed in the New York Herald of Feb. 28, without credit.

Erratum on page 258, 3d line from bottom. For 1000 meters read 1000

feet.

R. Rathbcn.

Fros. Amer. Phil. Soc. No. 10E.

Vol. XVIII, Plate VII.

--

F. Gutekunst.

Philadelphia.