The following is one of many ways in which this arrangement can be made, and it seems to be the simplest of them all.
Number the persons 1 to 8; and for our first day set them down in numerical order except that the two centre ones (4 and 5) change places:
| (1st day)— | 1 | 2 | 3 | 5 | 4 | 6 | 7 | 8 |
Keep the 1 and the 7 unaltered but double each of the other numbers. When the product is greater than 8, divide by 7, and only set down the remainder. Thus we get:
| (8th day)— | 1 | 4 | 6 | 3 | 8 | 5 | 7 | 2 |
(Here the fourth figure 3 is 5 × 2 ÷ 7, giving remainder 3, and so on.)
Repeat this operation once more:
| (15th day)— | 1 | 8 | 5 | 6 | 2 | 3 | 7 | 4 |
To fill in the intermediate days we have only to keep 1 unchanged and let the remaining numbers run downwards in simple numerical order, following 8 with 2, 2 with 3, and so on. Thus:—
| 1st day— | 1 | 2 | 3 | 5 | 4 | 6 | 7 | 8 |
| 2nd day— | 1 | 3 | 4 | 6 | 5 | 7 | 8 | 2 |
| 3rd day— | 1 | 4 | 5 | 7 | 6 | 8 | 2 | 3 |
| 4th day— | 1 | 5 | 6 | 8 | 7 | 2 | 3 | 4 |
| 5th day— | 1 | 6 | 7 | 2 | 8 | 3 | 4 | 5 |
| 6th day— | 1 | 7 | 8 | 3 | 2 | 4 | 5 | 6 |
| 7th day— | 1 | 8 | 2 | 4 | 3 | 5 | 6 | 7 |
| 8th day— | 1 | 4 | 6 | 3 | 8 | 5 | 7 | 2 |
| 9th day— | 1 | 5 | 7 | 4 | 2 | 6 | 8 | 3 |
| 10th day— | 1 | 6 | 8 | 5 | 3 | 7 | 2 | 4 |
| 11th day— | 1 | 7 | 2 | 6 | 4 | 8 | 3 | 5 |
| 12th day— | 1 | 8 | 3 | 7 | 5 | 2 | 4 | 6 |
| 13th day— | 1 | 2 | 4 | 8 | 6 | 3 | 5 | 7 |
| 14th day— | 1 | 3 | 5 | 2 | 7 | 4 | 6 | 8 |
| 15th day— | 1 | 8 | 5 | 6 | 2 | 3 | 7 | 4 |
| 16th day— | 1 | 2 | 6 | 7 | 3 | 4 | 8 | 5 |
| 17th day— | 1 | 3 | 7 | 8 | 4 | 5 | 2 | 6 |
| 18th day— | 1 | 4 | 8 | 2 | 5 | 6 | 3 | 7 |
| 19th day— | 1 | 5 | 2 | 3 | 6 | 7 | 4 | 8 |
| 20th day— | 1 | 6 | 3 | 4 | 7 | 8 | 5 | 2 |
| 21st day— | 1 | 7 | 4 | 5 | 8 | 2 | 6 | 3 |