# A walk of 3000 meters, but one foot has moved more, how so?

My math teacher has struck again.

Today I went for a normal walk of 3000 meters.
One of my feet had to move exactly 3000 meters.
However, the second foot moved 3100 meters.

Can you justify how did that happen?

### Edit:

I am not sure how this question has been selected as too broad.
It clearly only has a tag and the correct answer has been given and accepted in a fast way. If people decided to answer it using it doesn't mean that the question is too broad, it means that posters just want to get the "funny" comments and a "+1".

• Man, I wish this had a lateral thinking tag on it... I have this great answer about a guy attempting to step off a sidewalk and stepping back again numerous times due to traffic! Apr 27 '18 at 12:53
• I like how he seems to downplay his obsessive foot step measurement practices, causing all sorts of dysfunctions in his day-to-day life, by asserting that this is just “a normal walk” Apr 29 '18 at 0:00
• @MichaelK Do you recommend changing it to "ordinary" for example? to make it less ambiguious? May 2 '18 at 6:44
• No, actually, the choice of the word 'normal' more or less uniquely defines the correct solution to this riddle. May 2 '18 at 6:58
• @PaulKaram No, see my answer below: the word must be "normal" because that describes the walk very precisely; it makes the question a lot less ambiguous. That is the "lateral thinking" part of the question... to figure out that the word "normal" is not used in the... heh... normal sense it is used in common parlance. :) May 2 '18 at 7:12

and assuming his feet are 25 centimeters apart, if he walks $n$ circles with a radius of $r$ meters in a single direction, one foot walks $2 \pi r n$ meters, and the other one $2 \pi (r + 0.25) n$. The difference, $\pi n / 2$ is 100 meters, so $n = 200 / \pi \approx 63.6$. The value of $r$ is then determined from $2 \pi r n = 3000$ so $2 r = 15$ and $r$ = 7.5 meters.

Whether these

63.6 turns around e.g. a fountain or pond with a 15m diameter constitute a "normal walk" remains an open discussion.

• Very smart way of getting a variable $n$ into it :-) Apr 27 '18 at 12:57
• Oh, anything constituting $n$ and circles is normal for a Mathematician.
– Sid
Apr 27 '18 at 13:05
• It's not necessary for the path to be a uniform circle as long as the total amount turn is right. A 3 km straight walk where you make a little circle every 50 m (say, to look behind you) would also do. Apr 27 '18 at 15:06
• Does the same apply at certain latitudes simply by walking East-West? Apr 28 '18 at 20:32
• If he is positioned half-way between his feet (which sounds normal to me), won't he move 3050 metres while one foot moves 3000 metres and the other foot moves 3100 metres? Apr 30 '18 at 5:25

This person has an artificial left leg, and they were just minding their own business walking down the sidewalk when...WHAM!...a car comes along and takes out their left leg and sends it flying 100 meters!

• Wait this has gotta be the funniest answer I've seen in a long time :P +1 Apr 27 '18 at 15:17
• Thanks — Just had an out-loud laughing fit, at work, because of this answer :) Apr 27 '18 at 17:09
• Uh oh, funny answer. Counting down the minutes before a buzzkill mod with delete privileges comes around and obliterates it. Apr 27 '18 at 21:06
• Mod Buzzkill, reporting for duty. Actually this made me laugh out loud as well, and—in a rare fit of good humor—I'm pretending I didn't see this. Carry on!
– Rubio
Apr 29 '18 at 1:26
• The left leg doesn't have to be artificial. :P May 2 '18 at 5:53

You walked a full circle with diameter ~1000m

• I thought the same, however this would only work if his feet were ~16m apart... Apr 27 '18 at 12:46
• maybe they are do you have a problem with that? Apr 27 '18 at 12:47
• If pi * d = 3000, then assuming the two feet are 1 meter apart (an exaggeration, I know), then pi * (d + 1) = (pi * d) + pi, so the extra distance covered is only about 3 metres. Apr 27 '18 at 12:47
• Also see this puzzle which illustrates the same fallacy in another way. Apr 27 '18 at 12:49
• This is a good point @Phylyp, and seeing as this is tagged 'mathematics' I'm thinking I must be wrong Apr 27 '18 at 13:06

Well, I struggle a bit to explain my point, but

If you go back and forth on a straight line of 30 meters one hundred time and put the same foot half a meter meter ahead each time you reach the end of the line and have to turn around, your body will move 3000 meters, one of your foot will move 3000 and the other one 3100

• Step one meter ahead with one foot, then bring it back - it's moved two meters, not one. So with your example, the first time down the path you step 31 meters with one and 30 with the other. On every subsequent lap you're doing 32 and 30 (as one foot travels an extra meter at the start and end each time). After 100 laps your feet will have traveled 3199 and 3000 meters. You could adjust the number of laps you take the extra step... Apr 27 '18 at 14:07
• then go for 50cm instead of one meter :D not a typical walk but hey it fills criterias! Apr 27 '18 at 14:35
• The Ministry of Silly Walks would like to have a word Apr 28 '18 at 13:09

You were

running down a spiral staircase. this keeps one foot only mostly moving up and down, whiclst the outside foot also has circular movement. The circle is small enough that 100m is much more resonable

• You should hide your answer with the spoiler markup, and explain why you this this is the proper answer.
– Herb
Apr 27 '18 at 16:02
• Welcome to Puzzling.SE! Feel free to take the tour and visit the help center to learn more about the site. Apr 27 '18 at 16:13
• I like this answer, because it also incorporates the geometric meaning of "normal", ie. perpendicular to the plane of the ground. Apr 27 '18 at 21:12
• You should add an elevator every once in a while to reset your position, otherwise you may be on a nearly 1000 meters high staircase.
– Cœur
Apr 28 '18 at 12:26
• 300m of stair might be enough, still quite a workout. Apr 29 '18 at 9:42

Your professor went on a walk where he paced back and forth, as professors often do. If we assume the professor has a spacing of roughly 16 centimeters (~15.9154943) apart for his feet while walking, and that he always stops on the same foot, he will pivot on that foot, and need to swing his other foot in a half circle arc (Pi*Radius) before continuing in the other direction. At this spacing his second foot will move an extra .5 meters each time he switches directions. He will need to switch directions 200 times for the second foot to go 100 meters further. If we assume he doesn't pivot when starting, but pivots and turns one final time at the end before deciding to stop, he will need to pace back and forth a distance of 15 meters (3000/200) to have one foot go 3000 meters and the other 3100 meters.

• Welcome to Puzzling.SE! Make sure to take the tour! Apr 27 '18 at 20:15
• This answer has already been given. Apr 30 '18 at 22:44
• The closest answer has the person taking an extra step with one foot and then walking back, which is different from moving the foot in an arc at each turn around. May 1 '18 at 16:02

The amount of distance traveled by each foot is determined not only by the amount of distance it covers in the forward direction, but also the amount of distance it covers in the upwards and sideways directions. Most people don't have a perfectly symmetrical gait, so it's unlikely the exact path each foot takes on its way forward is the same. One foot probably steps a little higher or a little wider than the other, and over the course of a great many steps, these differences can add up. If one foot took a slightly longer step than the other, this would have an even greater effect. The foot with the shorter step would have to be raised more often for a given forward distance covered, and each time it is lifted, that adds a little upward distance to that foot's travel.

• I like the concept! It would work very well for someone who, say, couldn't bend one of his knees and has to swing that leg round. Apr 30 '18 at 9:21

# The key to the question is "Normal"

Today I went for a normal walk

A "normal" in geometry is not something "common" or "ordinary". A geometrical "normal" is always perpendicular to something else.

Let us say that this something is...

...a line that extends from a fixed point, to the walker.

If the walker then walks along the "normal" of this thing, they will...

...always walk in a circle. The point is the center of the circle. The line in question is the radius of the circle. Walking along the normal of a radius of a circle means you walk along the circle itself.

If you walk this way, one foot will move further than the other.

Now let us do the Maths:

Let us assume that the distance between the walker's feet are, well, one foot, 30 cm, or 0.3 meters. Let us assume they went $x$ number of laps around the circle. So the distances walked is $(r + 0) \cdot x \cdot 2\pi = 3000$ for the inner foot, and $(r + 0.3) \cdot x \cdot 2\pi = 3100$ for the outer.

Divide one by the other and we get:

$\frac{(r + 0) \cdot x \cdot 2\pi}{(r + 0.3) \cdot x \cdot 2\pi} = \frac{3000}{3100} \Rightarrow$

$x\cdot 2\pi$ cannot be zero so we can cancel that factor out

$\frac{r}{r + 0.3} = \frac{30}{31} \Rightarrow$

$31\cdot r = 30\cdot r + 9$

$r = 9$

Hence...

...they walked along a circle that had a radius of 9 meters, with the inner foot directly on the circle. I leave it to the reader to figure out how many laps it was.

• (But, other than specific figures, how is this different from the accepted answer?)
– Rubio
May 4 '18 at 14:51
• @Rubio The maths is the same but this answer comments on the "normal" part and explains why it must be a circle. May 4 '18 at 14:56
• I suppose. But the interpretation of "normal" in this sense isn't even necessary to arrive at this answer (the accepted answer didn't need to rely on it). Beyond that, the basic math is the same, and the substance of the answers is the same as well, making this answer really a dup with an extra interpretation tacked on that may or not have been intended by the OP. If all you are really adding is that interpretation, that would probably be better done by leaving a comment on the accepted answer.
– Rubio
May 4 '18 at 15:00
• It may not be necessary but it was included in the riddle, so I wanted to talk about why it is there. May 4 '18 at 15:13