Thursday, January 20, 2011

What's wrong with your head?

A fun puzzle from my son's friend. Thank you, Elliot!

A man set out for a walk. At the end of his walk his head had travelled 40 feet farther than his feet had travelled. He was a healthy man with all his limbs intact before and after the walk. So how did his head travel farther than his feet?

Answers accepted all day long on Friday, on our Family Puzzle Marathon. They will be hidden until Saturday morning (EST) and everyone who contributed something reasonable will get a puzzle point.

Image by darkpony. distributed under CCL.


SteveGoodman18 said...

There are a couple of possibilities...

First, the man could have fallen down about 4 times. For each fall, the head would travel to the ground and then back up as the man uprights himself again. His feet, however, would stay on or near the ground with each fall.

Second, the man could have been listening to some head-banging music and bobbing his head back and forth as he walked.

Third, the man might be the type who bows to everyone he meets as his way of greeting. Like the fall, the bow requires the man's head to lower and then raise, all while the feet stay on the ground.

Kim said...

He wasn't walking on level ground. When you walk on level ground, your head and feet travel the same distance, but when you walk over hills, your head actually travels farther.

Imagine, for example, walking all the way around one of those little planets in The Little Prince. In that case, your feet travel the circumference of the planet (2*pi*r), but your head travels in addition your height (2 * pi *(r+h))

In this case, if you went an extra 40', then:

2*pi*(r+h) - 2*pi*r = 40
2*pi*h = 40
h = 40/(2*pi) = 6'

Sounds about right!

Donna said...

My guess is that in addition to moving forward with the rest of his body, he was also turning his head from side to side and looking up and down as well.

Jenny said...

He was walking in a circle, leaning to the outside.

Dennis & Katrina said...

He was walking on the surface of a sphere. His feet will travel along the sphere's surface, traveling 2 * (pi) * r. His head, however, will travel 2 * (pi) * (r + his height). If we knew how tall he was, we could calculate the radius of the sphere.

This is true for us, walking on the Earth; however, that's a lot of walking. If you're 5' 10" and walking at the equator (the Earth's radius varies based on where you are, as it is not a perfect sphere), you'd have to walk completely around the Earth about 1.09 times for your head to have travelled 40 feet further than your head!

Wang said...

The only thing I can think of is another puzzle where it talks about a car having two wheels with much more wear than the other two wheels and it asks why that's the case.

The answer, for that puzzle at least, is that when a car turns, the outer two wheels travel more than the inner two wheels on the turn.

So similarly, in this case, if the head has traveled 40 feet farther, maybe he was walking on a curved surface?

For instance, if he walks on half a circle of radius 5 meters then his feet will have traveled 15.7 meters (5 * 3.14) but since his head is on the outer part of the circle (radius + his height, say 2 meters) then his head travels (7 *3.14) or 21.98 meters. His head will then have traveled 6.28 meters farther than his feet.

Tom said...

Unless it comes to me while I'm writing this, I'm stumped. And I assume it's not a trick question but it has a real math solution.
#1 idea: His head moves up and down as he walks, and his feet move up and down much less? (Nah. Feet likely elevate more than the head bobs.)
#2: His head moves side to side to side, and his feet wander less. Nah, not it.
#3: He's walking on a train, somehow? nah.

#4? He walked, like, around the world, or at least over a substantial mountain, and his own height made the circumference greater? Let's work with that...imagine he walks straight over a dome-shaped hill or mountain, the arc of which is one mile long, 5280' of foot travel. Assume the "center" of that arc is a mile underground (radius one mile). His head, perhaps 6' higher, is 5286' from the center, a bigger radius. (Draw the dome, and then draw the path of the head.) Seems like not enough, barely more than 1/10 of a percentage point difference. If this is a solution, and using these assumed numbers, his head MIGHT travel an extra 5.28 feet per mile of foot travel. An 8 mile hike over such a mountain (like 4 miles each way) could be what we're after here. Farther than I walk!

Still I don't like it; the head does travel largely "parallel" to the feet.

#5: I think I like #2 again, as the head may zigzag when we shift our weight. Probably hard to measure or calculate this, but that percentage difference may be greater than in #4. Can't wait to see others' solutions!

kj said...

We'll make some simple assumptions first:

The earth is a perfect sphere of radius r (in ft), and he is walking on a great circle (i.e. his path is on a circle of radius r).

If his height (in ft) is h, the top of his head has radius (r + h).

Now suppose he walks an arc with angle t (in radians). The arc length of his path for his feet is t*r ft, and for his head is t*(r + h) ft.

So his head travels t*(r + h) - t*r = t*h ft further than his feet.

For his head to travel 40 ft further than his head he would have to take a very long walk.

If he were 6 ft tall, that would mean t = 40/6 = 6 2/3 radians. One time around the earth is 2*pi radians = 6.2832 radians, so he has to walk around the earth more than once [(40/6)/(2*pi)= 1.061 times].

Peter Mesnik said...

I say his head traveled further than his feet because his head was rocking back and forth each time he took a step. This would increase the overall distance that his head would travel.

Ryan said...

He walked counterclockwise with his head tilted to the left (or clockwise with head tilted to the right), in a circle. So his head traced a circle with radius r1 and his feet traced a circle of radius r2. So:
2(pi)(r1)-2(pi)(r2)=40 thus

r1-r2= 40/(2pi) where r1-r2 would be the distance between his head and feet if his head were projected onto the ground (where the shadow would fall if the sun were directly overhead). There are infinitely many combinations that work, but only large values of r1 make for an anatomically correct person.

Cool stuff!

anne-marie said...

If a person walks at the equator, his/her feet would travel the distance 2PI*R.
R being the radiusof the earth.
However, his/herhead would travel 2 PI*(R+height of the person)
The difference between both would be 12PI feet if the height is 6.
In this case, this person is around 6.4 feet.(40/2PI).

Rabisabi said...

On behalf of one of my students:
Assuming his head is 1 foot ahead of his feet, as shown in the picture: every time he turns a corner, his head would travel 0.78 feet more than his feet. If he makes 50 turns, his head will have went 40 extra feet.

Andree said...

Right after he started out, 20 ft from his door, he thought he had forgotten something, his keys, perhaps, so in his head he retraced his steps, then figured out they were in his pocket. His head went an extra 40 ft.

Maria said...

We have so many amazing answers!
And they all very feasible.

Let me summarize:

1)Puzzle inventor's solution that many of you figured out. The man walked around the Earth, with his head making a circle of larger circumference than his feet.

2)Man had his headphones on and was shaking his head vigorously with the music, that after a few hours added up to 40 feet.

3)He has been falling down quite a lot, with his head going 4-5 feet up and down each time.

4)He has been bowing to every person he met.

5)Man had a big head and/or a big nose and has been turning corners frequently, with his head going on a slightly larger circle around the corner.

6)And the last, Hollywood-style solution, is that he had retraced his steps in his head without actually walking them.

I hope CIA has such a strong group of thinkers as all of you here. Another puzzle point for: SteveGoodman18, Kim, Donna, Jenny, Dennis & Katrina, Wang, Tom, kj, Peter, Ryan, anne-marie, Rob's student, Andree.

Some of you are turning 10 puzzles now and will be wrapped into a puzzle next week. Beware!

Maria said...

anne-marie and SteveGoodman18 - you each solved 5 puzzles now and qualify for a prize. Please, email me your mailing address and I will ship the prize to you.
maria at marialando dot com

Tom said...

Walked around the world. Right. Of course.

Just under 25000 miles, at the equator, 24,901. 40 miles less circumference if he did the polar route. Not counting mountains, and ignoring large bodies of water...

Hustling at 4mph, moving 12 hours a day, I get 518 days, roughly.

I like ALL the other answers better than that one.

Tom (later, Saturday evening)

anne-marie said...

It is a cool puzzle! We are going to think about it each time we walk now!

Lynnet said...

sorry I didn't post on friday, was busy.

he was on a space station and was walking around a circular object with a very small radius

Anonymous said...

he hunched.

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