A good method for solving problems is to break them down into smaller, easier problems.

Subroutine 1 (already explained by Neil)

First we focus on exchanging the places of just A and L, i.e. going from AL to LA.

L pulls A back, pushes it on the sidetracks and under the tunnel. L then disconnects, goes back to the main tracks, goes forward and returns to the sidetracks from the other side. It connects to A and pulls it on the main track. Voila! The order is now LA.

Now, we just use this subroutine (a computer term for a recipe) to solve the whole problem (we want to go from ALB to BLA). We have a subroutine that can exchange the places of a driving part and a non driving part (i.e. we can go from ALB to LAB or to ABL, but cannot go from AB to BA because neither A nor B is a driving part!).

The solution consists of taking AL as the driving part and B as the non-driving part. We can go from (AL)B to B(AL). Now we take L as the driving part and A as the non-driving part, going from BAL to BLA. We are done!

1) A road and a railroad run parallel to each other. A man cycles to work every morning at a constant speed of 12 M.P.H. Normally, a train traveling at an unknown constant speed in the same direction passes him exactly in front of 2 very large oak trees. One day, the man was 25 minutes late for work and the train, always on time, passed him 6 miles before the oak trees. What is the speed of the train?

2) If a girl takes 3 steps to a man's 2 steps and they both start on the left foot, how many steps do they have to take before they are both stepping on the right foot together?

3) A man gets out of his house, walks South for a mile, then East for a mile. He spots a bear and shoots but misses. He goes North for a mile and finds himself home. What color is the bear?

4) If we place 4 pencils as if we were going to play tic-tac-toe, we have 16 right angles.

How can we place 3 pencils so as to get 12 right angles?

5) You have a barrel containing 8 gallons of beer and 2 empty jugs, one that will hold 5 gallons and one that will hold 3 gallons. How do you divide the beer equally between 2 men (4 gallons each) without spilling any or using any other receptacle? Hint: We start at 800, go to 350 to 323…we want to end up at 440.

1) Let's assume that the train passes the man each day at 9 a.m. in front of the oaks. Today the man will reach the oaks at 9:25 a.m.. Since today the train passes him 6 miles before the oaks and since he travels at 12 M.P.H., he is passed by the train half an hour before 9:25, i.e. at 8:55. The train has exactly 5 minutes to reach the oaks at its usual 9 a.m., hence the train travels 6 miles in 5 minutes, or 72 miles in 60 minutes.

2) Let's make a drawing of the steps:

If you notice that where the * is, the pattern starts repeating itself, you can tell right away that they will never take a right step together.

3) Where in the world can you make a U shape and get back where you started? Let's look at our earth more carefully: the east-west lines (including the equator) are called latitudes. The north-south lines are called longitudes and they seem parallel to us, but they do meet at the poles! The only places where our man can live are at the North Pole or near the South Pole: the bear had to be white.

4) The first 2 pencils must cross each other, making 4 right angles. How can we place the third pencil to make 8 more? The solution cannot be drawn in the plane of the first 2 pencils: you have to think in 3 dimensions and the answer becomes obvious. The third pencil sticks straight out of the plane and forms 4 right angles with each of the first 2 pencils.

5) We want to go from 800 to 440. It is easy to go from 800 to 350 to 323 to 620. But you could also go from 800 to 053 or from 800 to 503. Not knowing what to do next, I decided to work backwards from 440. Before that, I had to come from 143, before that from 152, before that from 602, before that from 620. Aha! I already know how to get to 620, so I am done!

My factory has 10 machines which all produce widgets. All the widgets weigh 23 grams. Now one of the machines is broken and has been producing widgets that are all identical but either heavier or lighter than 23 grams. How can I tell which machine is broken by using my excellent scale only twice!

Hint: First, find the amount by which the faulty widget differs from 23 grams.

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