THE STAIRCASE RACE

       This is a rough sketch of the finish of a race up a staircase in which three men took part. Ackworth, who is leading, went up three steps at a time, as arranged; Barnden, the second man, the second man, went four steps at a time, and Croft, who is last, went five a time. Undoubtedly Ackworth wins. But the point is, how many steps are there in the stairs, counting the top landing as a step?

        I have only shown the top of the stairs. There may be scores, or hundreds, of steps below the line. It was not necessary to draw them, as I only wanted to show the finish. But it is possible to tell from the evidence the fewest possible steps in that staircase. Can you do it?

                                                          TIMING THE CAR

        “I was walking along the road at three and a half miles an hour,” said Mr. Pipkins , “ when the car dashed past me and only missed me by a few inches.”

         “Do you know at what speed it was going?” asked his friend.

         “Well, from the moment it passed me to its disappearance round a corner

I took twenty-seven steps and walking on reached that corner with one hundred and thirty-five steps more.”

          “Then, assuming that you walked, and the car ran, each at a uniform rate, we can easily work out the speed.”

                                                          SHARING A BICYCLE

             Two brothers had to go on a journey and arrive at same time. They had only a single bicycle, which they rode in turns, each rider leaving it in the hedge when he dismounted for the one walking behind to pick up, and walking ahead himself, to be again overtaken. What was their best way of arranging their distances? As their walking and riding speeds were the same, it is extremely easy. Simply divide the route into any even number of equal stages and drop the bicycle at every stage, using the cyclometer. Each man would then walk half way and ride half way.

              But here is a case that will require a little more thought. Anderson and Brown have to go twenty miles and arrive at exactly the same time. They have only one bicycle. Anderson can only walk four miles an hour, while Brown can walk five miles an hour, but Anderson can ride ten miles an hour to Brown’s eight miles an hour.

               How are they to arrange the journey? Each man always either walks or rides at the speeds mentioned, without any rests?