The content of this article is excerpted from "Fascinating Logic Problems" by [English] alex bells, translated by Hu Xiaorui, published by CITIC Publishing House in October 2018, with deletions, please indicate the source.
In an early Sherlock Holmes series, The Adventure of the Priory School, Holmes deduced the direction of the bicycle from the mere two wheel marks left by the bicycle. He explained his reasoning process to Watson: "The weight of the bike all falls on the rear wheels, so the wheel marks left by the rear wheels will of course be deeper. You see, there are a few places where deep wheel prints run over shallow tops and completely overshadow shallow wheel prints. There is no doubt that the bike is leaving the monastery. ”
I don't quite understand his reasoning. But no matter which direction the cyclist goes, the rear wheels will cover the marks left by the front wheels, won't they?
It is indeed possible to infer the direction of the bicycle using wheel marks, but Sir Arthur Conan Doyle, the author of Sherlock Holmes, missed an opportunity for self-expression.
Puzzles for bicycles
01
The picture below is the wheel mark left by the cyclist, is his direction of travel from left to right or from right to left?
Figure 1
Holmes is right, you first have to distinguish which wheel print is left by the front wheel and which one is left by the rear wheel. However, you don't need to know the depth of the wheel print to get the job done.
To deduce the direction of travel of the bicycle, we must first determine which wheel print is left by the front wheel and which one is left by the rear wheel.
If the wheel print is straight, the direction of travel of the wheel is consistent with the direction of the wheel print. However, if the wheel print is curved, the direction of travel of the wheel will be consistent with the tangent direction of each point on the wheel print. (A tangent is a line that has only one point of contact with a curve.) For ease of understanding, take a closer look at the wheel marks left by the unicycle in Figure 2 below. The orientation of the wheels at points a, b, and c is the same as the direction of each tangent I marked.
Figure 2
The bicycle has two wheels, the front wheel can point in any direction, and the rear wheel has no freedom in the direction of travel - it must always point in the direction of travel of the front wheel.
Therefore, no matter where the rear wheel is located, the front wheel is in the tangent direction of the rear wheel print, and the distance from the rear wheel must be equal to the length of the bicycle's body. In other words, all tangents of the rear wheel print must intersect the front wheel print, and the distance between the cut point and the intersection point must be equal to the length of the body.
Now, please look at the dot d on the thick line in the figure below (Figure 3). The tangent at this point does not intersect the two wheel prints. Therefore, we can conclude that the curve where the fixed point d is located is not the rear wheel print, but the front wheel print.
Figure 3
Now we can determine the direction of travel of the bicycle. We know which wheel print is left by the rear wheel, and the discussion above also tells us that along any tangent line on the rear wheel print, the length of a bicycle's body in the direction of the bicycle's travel will be the same as the front wheel print
Intersect at a certain point. Therefore, we take two points e and f on the rear wheel print, draw the tangent lines of the rear wheel print respectively, and observe the intersection of these two tangent lines with the front wheel print. From Figure 4, it can be seen that the two segments on the left side of points e and f are equal in length, while the two segments on the right are not equal in length. The distance between the wheels does not change during travel, so it can be seen that the direction of travel of the bicycle is from right to left. It's so simple!
Figure 4
02
A photographer photographs a bike in motion. The bike travels along a horizontal road, but we don't know if it's going in the direction from left to right or right to left. But the direction doesn't matter. The bike's wheels are disc-shaped with two pentagonal signs on it. Which of the two graphics below is a photograph taken by a photographer?
Figure 5
I particularly like this question because it reflects a strange phenomenon: the top of the wheel always runs faster than the bottom.
When the wheel rolls along the horizontal plane, the points on the wheel need to complete the movement in two different directions. They need to move horizontally in the direction of the wheel; in addition, they need to rotate around the center of the wheel. Movements in both directions sometimes complement each other and sometimes cancel each other out. We take any point on the wheel. As shown in the figure below (Figure 6), when the point is at the top of the wheel (at point a), the horizontal motion and rotational motion of the point complement each other. However, when this point is at the bottom of the wheel (at point b), the direction of the horizontal motion and the rotational motion is exactly the opposite and cancels out each other. From the observer's point of view, when the wheel rolls, the speed of the top is always twice the horizontal speed of the wheel, while the bottom of the wheel is always stationary. It can be seen that the movement speed of the points in the lower half of the wheel is slower than the points in the upper half.
Figure 6
Therefore, the correct answer is the second picture. Because in this picture, the upper pentagon is blurred, while the lower pentagon is very clear. If the photographer sets the exposure time of the camera short enough, the slower pentagon will leave a sharp image, but the faster pentagon will definitely be blurrier. If you like to draw, you may understand the reasoning right away. When drawing moving wheels, the top is often blurred.
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