Posted February 2, 2010 By Presh Talwalkar. Read about me , or email me .



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"All will be well if you use your mind for your decisions, and mind only your decisions." Since 2007, I have devoted my life to sharing the joy of game theory and mathematics. MindYourDecisions now has over 1,000 free articles with no ads thanks to community support! Help out and get early access to posts with a pledge on Patreon. .

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Consider the picture below of bicycle tracks in the mud.

Can you figure out which way the rider was going, to the left or to the right?

This is not an easy puzzle. In fact, it was such a puzzle that once befuddled the great detective Sherlock Holmes.

But the picture provides just enough clues to be solved. Let us investigate the mystery.

Sherlock Holmes’ mistake

The background to this puzzle is the Sherlock Holmes tale The Adventure of the Priory School by Sir Arthur Conan Doyle.

At one point in the story, Holmes and Watson approach a set of bicycle tracks left in mud.

What can be deduced? Here is how Holmes reads the scene:

“This track, as you perceive, was made by a rider who was going from the direction of the school.” “Or towards it?” “No, no, my dear Watson. The more deeply sunk impression is, of course, the hind wheel, upon which the weight rests. You perceive several places where it has passed across and obliterated the more shallow mark of the front one. It was undoubtedly heading away from the school.”

Holmes unsurprising arrives at the right answer in the story version. But this was somewhat lucky because his logic is shaky.

The suspect part is the bit about the hind wheel obliterating the mark of the front wheel. This alone actually gives no clue about the bike’s direction.

Why? It’s because the hind wheel always follows the front wheel because it cannot turn. The hind wheel will always obliterate the front wheel marks, regardless of whether the rider was heading toward or away from the school. Nothing can be inferred on this basis alone.

Holmes was on the right track, if you will, but he did not go far enough. To solve this mystery correctly, we will take a step back for a moment.

Re-creating the scene of the crime



How can we figure out the direction of the bike rider from the track marks?

Think for a moment about how the marks are created. A rider gets on a bike, pedals to move the hind wheel, and steers by moving the front wheel. The front wheel makes the leading mark and the hind wheel follows in the same direction.

What we need to do is reverse engineer this process. We have tire marks and we need to identify which track belongs to which wheel, and more importantly, which direction the bike was moving.

And we will rely on two key facts to answer these questions:

Fact 1: the hind wheel moves in the direction of the front wheel

Fact 2: the hind wheel is a fixed distance from the front wheel

The first fact is a consequence of the hind wheel not being able to turn. The hind wheel is fixed by the bike frame in a single direction, and hence it always moves to follow the direction of the front wheel.

The second fact is a consequence of a bike design which fixes the centers of both the front and hind wheel. The hind and front wheels’ point of contact with the ground is equal to some fixed length.

So with this understanding, how can we identify the tracks and the direction of the bike?

A small and useful tangent



We will use an assist from physics (or calculus) to solve this mystery. If you’re familiar with projectile motion you can skip this. Otherwise I will give a brief refresher.

The idea we need has to do with modeling the motion of on a curve. A common motivating example is to consider lobbing a baseball to your friend. The ball leaves your hand with some speed and it gently rises until its maximum before falling in a symmetric curve towards your friend. The shape of the curve is a parabola and this is a canonical example of projectile motion.

While we know the path of the ball, we may wish to dig deeper. One might ask: what is the direction of the ball at any given moment?Â To figure this out, you might first draw the entire path of the ball, and then you can approximate. What you do is draw a line that connects a given position with a future position. This gives an approximation to the ball’s direction. The approximation can get better by decreasing the increment. And using calculus it’s possible to calculate the limit of this process, and the resulting line is known as the tangent line, the line that just kisses the curve:

The important fact is that the tangent line at a point indicates the direction of the ball at that point on the curve. More generally, on a curve, the tangent line at a point indicates the direction of the curve at that point. And this is the key to solving the bicycle mystery.

A method to solve the mystery



Let us put this all together.

We have two sets of bicycle tracks. We know the curve for the hind wheel is “following” the curve for the front wheel for the rider’s direction, but we cannot identify the curves nor can we identify the direction.

So what we will do is investigate. We take several points on both curves and we will draw tangent lines. We do not know which direction the tangent line should go (since we don’t know which way the bike was moving), so what we will do is draw the tangent line in both directions.

We know have tangent lines on two curves going in two different directions (4 possible candidates).

We will be able to identify the hind wheel’s marks going in the correct direction because it will have the following characteristics:

By fact 1, the tangent line will intersect the other curve that represents the front wheel marks

By fact 2, the tangent line in the correct direction will intersect at a fixed distance

We will consequently have our answer of which way the bike was going!

To recap, here is the process we will use:

–Choose several points on each curve

–Draw the tangent lines going in both directions for these points

–Identify which set of tangent lines best matches the characteristics of the hind wheel moving in the correct direction

This is a lot of words but it will make more sense visually, as pictured below.

Investigating the tire marks!



I have drawn out my work below.

What I have done is chosen several points, drawn tangent lines in both directions, and then color-coded them for ease.

Notice the blue tangent lines are all over the place. They do not intersect with the other curve and hence we can conclude this is the path of the front wheel.

The green tangent lines, on the other hand, are confined. They do intersect the other curve at a regular fixed length to their right. Thus we can conclude this is the path of the hind wheel going right.

(And I admit, the picture isn’t perfect because I generated these lines by sketching, but you get the idea.)

We can thus identify the path of the hind wheel, and we can conclude the bike must have been moving to the right.

Pretty cool, isn’t it?

Pat yourself on the back–you are now smarter (about this problem) than Sherlock Holmes.

References:

Edward Bender’s paper about Sherlock Holmes

Discussion Questions:

1. Ride your bide in the mud. Or take a look at a picture of tire marks. What other clue is there that we didn’t have?

2. Would the exercise change for a motorcycle? What about a car?

3. Would it matter if a car was front-wheel or rear-wheel drive?