A mystery Sherlock Holmes couldn’t solve, but you can

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?  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?



Share this post:

| More

Previous post:

Next post:



  • danihel

    Hi!
    I’ve been reading your blog for some time now (since the Dark Knight article) and I find it very useful. I keep coming back because some articles really make me think.

    To the discussion questions:
    1. The back wheel track would step on the front wheel track making things easier.
    2. Things wouldn’t change for a motorcycle. For the car you will have to consider a single side of the car.
    3. It doesn’t matter for the traction, I think it is the same thing.

    I’m not sure about the second question. Is it ok?

  • E

    Can anyone confirm that this works in real life? I imagine that the spacing between the two tracks is generally so small that only in the sharpest of corners can you tell which wheel is in back and rarely get the direction.

    Bike tread is usually shaped like chevrons to provide traction in the mud or rain. If the tires were put on correctly, the chevrons in the mud will usually point to the source of the bike ride.

  • felix

    I figured the bike was riding to the left because if it was going from left to right, why would the back wheel go past the front wheel on the first slope right after the first intersection of the tracks, i.e. why would the tangent of the back wheel be not as steep as the tangent of the front wheel. This, however, is only a little flaw in the drawing and the evidence you pointed out is greater :)

  • http://www.brelip.com Duo

    Hmm… interesting problem. However I still think it is insufficient to conclude about the direction.

    According to your reasoning, you can also conclude bike is going to the left – you can draw equi-distant tangent to the left as well.

    I agree with the wheels assessment though since front is always leading.

  • imblo

    Duo – the distance from the tangent point of the rear wheel curve to where it intersects the front wheel curve on the left hand side changes, unlike going to the right.

    If the bike were going to the left, this suggests that the distance between the rear and front wheels is changing, which cannot be.

  • http://dewimorgan.livejournal.com Dewi Morgan

    1 As well as the clue mentioned by danihel (and Holmes) that the rear wheel would cross the tracks of the front wheel, there would also be splatter and skid.

    2 Not significantly.

    3 Front/rear wheel drive, no. Front/raer wheel STEERING, yes.

  • http://phillipsjk.ca James Phillips

    From looking at the diagram, I concluded the bicycle must be moving to the left. I had this gut feeling that a rider going the other way would be more likely to fall over. A consequence of being a sketch I suppose.

    As well, when the tracks intersect on the left hand side, the tangent from the rear wheel is *not* immediately intersecting the “leading track” toward the right. It intersects sooner toward the left.

    The concept of balance brings up an interesting question: can you determine the speed of a cyclist by analyzing their wobble? The faster you go, the more “lean” you need for turns. I suspect you would have to know the length of the wheel-base, but you claim that can be determined with tangent lines.

  • Pingback: Logic Questions - All The Misc

  • http://www.mindyourdecisions.com/blog/ Presh Talwalkar

    Thanks for all the comments. Chevrons can help though not all bike treads have them, like, say, a street bike.

    I did my best with the drawing…if I find a better way to draw it I will update the pictures.

  • Sat

    Well there is nothing wrong with the picture… in real life any combination of curves are possible depending on (a) terrain, (b) speed of the bi-cycle, (c) skill level of the rider.

    The fact is unless we have atleast 2 sharp curves it will be difficult to assess the direction. Lets say, a very skilled rider going very fast on a smooth terrain will produce no curves and hence no way to judge the direction.

  • http://phillipsjk.ca James Phillips

    The other day I got a very good tack in snow, but when I went back the next morning it was gone (they may have groomed the trail).

    As I was moving at low speed, I had a very pronounced front wheel wobble, but the rear wheel showed no noticeable deflection: that is to say the tangent from the rear wheel did not always intersect the track of the front wheel.

    I don’t think the distance between the front and rear wheels was changing noticeably: though the speed of rotation of the two wheels relative to each other may have been changing. The wobble may have been timed to match my down-strokes on the pedals. My winter tires do have chevrons, but it is trivial to install the tire backwards! In snow or mud slippage is probably the best indication of direction: The “knobs” (whether directional or not) on the tires will make a track the “points” (or drifts) in the direction of movement (during side to side slipping). Wheel spin will deposit debris behind the point where the tire spins (which shows up as a smooth spot).

    A better way to draw the lines would be to run the wheels of a model bicycle through ink, then run that across a sheet of paper. The problem is that such a model would not balance like a real rider. Then there is the more tricky and bandwidth hungry method of photographing actual tracks.

  • Someone

    The easy way to discriminate front from back wheel is to notice that the back wheel always draws a shorter path than the front wheel.

  • john

    Good analysis, except for 2 minor points.

    First, the rear tire will always be on the inside of any substantial curve. Your tangents not crossing the other line is a way to demonstrate it. I just think it’s simpler to point out that the rear will be on the inside (just a personal preference probably).

    I used the same method as you and decided the lines aren’t well drawn, as the tangent distance from the rear to the front is not constant in either direction. It appears to me that the distance is closer to constant going left than going right.

    Your conclusion comes from the fact that you only use 3 points. Using almost any point on the near vertical part on the left would show that going right cannot work. Also, if going right, the inflection point of the rear wheel curve is after it crosses the front wheel’s track. Taking your tangents between the crossing and the inflection point, you would have to conclude that the front wheel was going left as the rear wheel went right.

    Your analysis would be good if you knew that the tracks were actually made by a bike, but these tracks were more likely made by 2 unicycles. They are actually more likely an artist’s rendition, and are not a depiction of real tracks at all.

    Non the less, you did find the correct method to deduce direction where Sherlock Holmes failed.

    Congratulations

  • Pingback: One mile south, one mile east, one mile north – a classic puzzle - Mind Your Decisions

Previous post:

Next post:

This site is for recreational and educational purposes only (privacy policy)

Powered by WordPress