No. LXVIII.—TO COLOUR MAPS
Four colours at most are needed to distinguish the surfaces of separate districts on any plane map, so that no two with a common boundary are tinted alike.
On this diagram A, B, and C, are adjoining districts, on a plane surface, and X borders, in one way or another, upon each.
It is clearly impossible to introduce a fifth area which shall so adjoin these four districts as to need another tint.
A FREAK OF FIGURES
Here is another freak of figures:—
| 9 | × | 1 | - | 1 | = | 8 |
| 9 | × | 21 | - | 1 | = | 188 |
| 9 | × | 321 | - | 1 | = | 2888 |
| 9 | × | 4321 | - | 1 | = | 38888 |
| 9 | × | 54321 | - | 1 | = | 488888 |
| 9 | × | 654321 | - | 1 | = | 5888888 |
| 9 | × | 7654321 | - | 1 | = | 68888888 |
| 9 | × | 87654321 | - | 1 | = | 788888888 |
| 9 | × | 987654321 | - | 1 | = | 8888888888 |
No. LXIX.—THE TETHERED BIRD
A bird made fast to a pole six inches in diameter by a cord fifty feet long, in its flight first uncoils the cord, keeping it always taut, and then recoils it in the reverse direction, rewinding the coils close together. If it starts with the cord fully coiled, and continues its flight until it brings up against the pole, how far does it fly in its double course?