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My math teacher has struck again.

Here's his newest riddle:

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".

Community
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Paul Karam
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    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! – Phylyp Apr 27 '18 at 12:53
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    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” – kedarguru Apr 29 '18 at 00:00
  • @MichaelK Do you recommend changing it to "ordinary" for example? to make it less ambiguious? – Paul Karam May 02 '18 at 06:44
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    No, actually, the choice of the word 'normal' more or less uniquely defines the correct solution to this riddle. – Glorfindel May 02 '18 at 06:58
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    @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. :) – MichaelK May 02 '18 at 07:12
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    The excessive answers, many of which are definitely (and even admittedly) [tag:lateral-thinking] despite the question being solidly grounded on [tag:mathematics], are inappropriate. If you're not making a good faith effort to answer the question as posed, please restrain your urge to indulge in posting "clever", "funny" or "unique" solutions. We routinely delete responses that rely on lateral thinking when no such tag was indicated, as they are Not an Answer to the actual puzzle. Most of these answers are likewise Not an Answer. They will be deleted soon. – Rubio May 03 '18 at 21:13

8 Answers8

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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!

The Netty Professor
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    Wait this has gotta be the funniest answer I've seen in a long time :P +1 – NL628 Apr 27 '18 at 15:17
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    Thanks — Just had an out-loud laughing fit, at work, because of this answer :) – Caleb Apr 27 '18 at 17:09
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    Uh oh, funny answer. Counting down the minutes before a buzzkill mod with delete privileges comes around and obliterates it. – user1717828 Apr 27 '18 at 21:06
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    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 01:26
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    The left leg doesn't have to be artificial. :P – JRN May 02 '18 at 05:53
  • Gotta be at the very end of that wonderful morning walk. – Adnan Khan May 02 '18 at 08:51
  • I think it would have to go half that and then be returned. Although this does contradict the "normal" part. – jpmc26 May 03 '18 at 10:27
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    @JoelReyesNoche your comment and this answer is hilarious! Favourited :D $\color{darkorange}{\bigstar}$ – Mr Pie Aug 22 '18 at 12:20
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Based on @TwoBitOperation's answer

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.

Robert Longson
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Glorfindel
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    Very smart way of getting a variable $n$ into it :-) – Phylyp Apr 27 '18 at 12:57
  • Well, the normal is just to say that he's walking straight, there's no lateral thinking or a weird way of moving. Well, the answer is weird, and you hit it better than @TwoBitOperation since it's more logical, but it's still a bit "normal" :p – Paul Karam Apr 27 '18 at 13:01
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    Oh, anything constituting $n$ and circles is normal for a Mathematician. – Sid Apr 27 '18 at 13:05
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    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. – hmakholm left over Monica Apr 27 '18 at 15:06
  • How far do you need to alter the distance between feet to get a whole number of circles? – Joshua Apr 27 '18 at 21:46
  • I note that it is frequently easy to walk in squares with monotonic winding number, especially in cities. – Eric Towers Apr 28 '18 at 02:46
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    Does the same apply at certain latitudes simply by walking East-West? – Sentinel Apr 28 '18 at 20:32
  • perhaps he's turning the handle on a large machine? – Jasen Apr 29 '18 at 10:04
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    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? – CJ Dennis Apr 30 '18 at 05:25
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    @Sentinel If one is 7.5m from the North or South poles. – Acccumulation Apr 30 '18 at 18:24
  • Yes, but it would still 'feel' as a rather small circle. – Glorfindel Apr 30 '18 at 20:46
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    @CJDennis good point. I'm not sure how marathon tracks (or any athletic outdoor races) are measured, it could very well be the 'inside' of the track. Especially when there are a lot of curves, you'll end up running more (3050m measured from the center of your body) than the indicated distance (3000m). – Glorfindel Apr 30 '18 at 20:49
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    @Glorfindel I think it is the inside as the athletes hug the inside of the lane to reduce the distance. That's also why the starting positions are staggered when the race has curves, so that every athlete runs the same distance. – CJ Dennis Apr 30 '18 at 22:50
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    "Normal" here has nothing to do with "ordinary" or "common". A mathematical normal is a direction, at a right angle to something. So, imagine there is a line from a point to the walker, and have the walker always move along the normal of that line, and then the walker will automatically walk in a circle. – MichaelK May 02 '18 at 06:41
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You walked a full circle with diameter ~1000m

TwoBitOperation
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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

RiddlerNewComer
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  • 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... – sirjonsnow Apr 27 '18 at 14:07
  • then go for 50cm instead of one meter :D not a typical walk but hey it fills criterias! – RiddlerNewComer Apr 27 '18 at 14:35
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    The Ministry of Silly Walks would like to have a word – nzaman Apr 28 '18 at 13:09
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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

Zarwalski
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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.

A. Clark
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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.

Lily Finley
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  • 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. – AndyT Apr 30 '18 at 09:21
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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.

Rubio
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MichaelK
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    (But, other than specific figures, how is this different from the accepted answer?) – Rubio May 04 '18 at 14:51
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    @Rubio The maths is the same but this answer comments on the "normal" part and explains why it must be a circle. – MichaelK May 04 '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 04 '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. – MichaelK May 04 '18 at 15:13