asionally come across remarkable
coincidences--little things against the probability of the occurrence of
which the odds are immense--that fill us with bewilderment. One of the
three motor men in the illustration has just happened on one of these
queer coincidences. He is pointing out to his two friends that the three
numbers on their cars contain all the figures 1 to 9 and 0, and, what is
more remarkable, that if the numbers on the first and second cars are
multiplied together they will make the number on the third car. That is,
78, 345, and 26,910 contain all the ten figures, and 78 multiplied by 345
makes 26,910. Now, the reader will be able to find many similar sets of
numbers of two, three, and five figures respectively that have the same
peculiarity. But there is one set, and one only, in which the numbers
have this additional peculiarity--that the second number is a multiple of
the first. In other words, if 345 could be divided by 78 without a
remainder, the numbers on the cars would themselves fulfil this extra
condition. What are the three numbers that we want? Remember that they
must have two, three, and five figures respectively.

102.--_A Reversible Magic Square._

Can you construct a square of sixteen different numbers so that it shall
be magic (that is, adding up alike in the four rows, four columns, and
two diagonals), whether you turn the diagram upside down or not? You must
not use a 3, 4, or 5, as these figures will not reverse; but a 6 may
become a 9 when reversed, a 9 a 6, a 7 a 2, and a 2 a 7. The 1, 8, and 0
will read the same both ways. Remember that the constant must not be
changed by the reversal.

103.--_The Tube Railway._

[Illustration]

The above diagram is the plan of an underground railway. The fare is
uniform for any distance, so long as you do not go twice along any
portion of the line during the same journey. Now a certain passenger,
with plenty of time on his hands, goes daily from A to F. How many
different routes are there from which he may select? For example, he can
take the short direct route, A, B, C, D, E, F, in a straight line; or he
can go one of the long routes, such as A, B, D, C, B, C, E, D, E, F. It
will be noted that he has optional lines between certain stations, and
his selections of these lead to variations of the complete route. Many
readers will find it a very perplexing little problem, though its
conditions are so simple.

104.--_The Skipper