On Friday I posted this puzzle…..

Imagine that you are standing on the banks of a straight river. Across from you is a sign on the opposite bank. To your right is a tree. You have nothing to measure with and cannot cross the river. How can you figure out roughly how far you are from the sign?

If you have not tried to solve it, have a go now. For everyone else the answer is after the break.

Walk along the river to the tree, counting your steps as you go. Now count out the same number of steps beyond the tree, turn right by 90 degrees, and walk away from the river until you are in line with the sign and tree (see diagram), counting your steps as you walk. Multiply the number of steps by the distance of a single step, and you will have calculated the rough distance between your original position and the sign. Did you solve it? Any other answers?

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My solution too – but I imagine there will be others. Also, this only works if the river has a straight bank of course.

Actually, to mildly correct myself, the river bank doesn’t have to be straight. All that’s required is to be able to walk in a straight line beyond the tree, so it could bend upwards relative to the diagram.

I tried that, but the fox ate my chicken.

*like*

I had that you could walk to the tree, estimate the angle between the bank and the line to the sign, and pull some trigonometry.

Same here, but I wasn’t very happy with it since I don’t usually carry with me instruments for measuring angles. Unfortunately I couldn’t think of Richard’s solutions. But arguably his solution also requires a reliable way to measure 90°.

Fist at arms length covers about ten degrees. Easy as pie.

I have a better answer.

Walk along the river to the tree, counting your steps as you go. Now count out the same number of steps beyond the tree, turn right by 90 degrees, and walk away from the river until you are in line with the sign and tree, counting your steps as you walk. Multiply the number of steps by the distance of a single step, and you will have calculated the rough distance between your original position and the sign.

Eh? This is Richards solution?

Eh? (seconded)

I think mine works slightly better:

Walk along the river to the tree, counting your paces as you go. Now count out the same number of paces beyond the tree, turn right by 90 degrees, and walk away from the river until you are in line with the sign and tree, counting your paces as you walk. Multiply the number of paces by the distance of a single pace, and you will have calculated the rough distance between your original position and the sign.

Paces are more accurate than steps.

Good one … better than mine

… much more pertinent given that Steps are making a comeback

Lol. Nice one.

I assumed that counting steps was a way of measuring and dropped all solutions which involved it. Bad phrasing of the question, Dr. Wiseman.

A step becomes a unit and my foot a measuring rod.

My rod is a foot (?!?!?)

I like the solution and didn’t think of it.

However, this part “Multiply the number of steps by the distance of a single step” ok, so, how do we know the distance of a single step?

Anyway that’s just a wording problem. Could just as easily have said “you only have a 1m tape measure” or similar.

You know your own length, so you can also compare your own length with the size of a step. There are ways to do that.

I suppose the way around it would be to approximate the size of your foot as 30cm, so you could either ‘measure’ the distance by stepping with one foot directly in front of the other, or if you could somehow mark the start and end of a pace…

I thought you walk in the direction of the tree, until the angle between your starting position (you marked in the dirt) and the sign is 45 degrees (you estimate the angle by bisecting lines in the sand). However far you’ve just walked, that’s your distance.

The tree isn’t strictly necessary in my solution, but it helps you walk straight.

If you can see a mark in the sand from afar, you don’t need the tree even in Richard’s solution….

No need to go so far. Just count steps to the tree, divide by 4 then multiply by 3. It’s a right angled triangle, or 3, 4, 5 triangle.

That would only work if the sign is 3/4 of the distance from you that the tree is. The tree could be twice as far away and it would still be a right angled triangle.

Good puzzle, by the way. I didn’t get the solution on Friday.

I had two sollutions with angles, but this one is much more elegant and precise.

You can also wait until it starts freezing, and walk to the sign, count your steps as you go.

Like Navneeth I ignored my bodies physical capabilities of measurement. I did however chop down the tree, place it carefully over the river. If it fitted, the sign would be about a tree length away. Could scan brain for all tree knowledge on average lengths of species etc.

The answer reminds me of the box you need to draw using 3 lines and go outside the box to complete it. Or whatever the thingy is ; )

I just paced it out, straight line.

I shoud have just asked you.

I assumed that “nothing to measure with” didn’t involve my dismemberment, that the river banks were flat and I got it.

I had considered standing next to the tree and judging the visual size of the sign (e.g. by holding up a thumb and comparing apparent relative sizes), then reversing away from the sign keeping it in line with the tree, until its apparent size had halved. At this point you know you are twice as far from the sign as you were … e.g. the distance you have walked from the tree equals the distance of the tree from the sign. At this point you then know 2 sides of a right angle triangle and can calculate the river width from Pythagoras. Phew. Richard’s solution is much easier and more elegant!

My solution was to count my steps as I walked west (relative to the picture) and then counted my steps along the end of the river and then counted my steps to the sign and subtract out my east and west steps.

The three points form a triangle with one 90 degree angle, roughly the angle at the tree 45 degree. Thus the distance between you and the tree roughly equals the distance between me and the sign.

Actually, mine was slightly different and based on the procedure (or so I’ve heard) used by Napoleon’s troops.

Stand opposite the sign (position 1) and bring your hands up to your eyebrow (like shading your eyes from the sun)

Tilt the hand down until the bottom edge intersects with the BASE of the sign.

Keeping the hand in the same position, rotate your body 90 degrees toward the tree.

Walk forward or backward as needed until the edge of the hand intersects the base of the tree (position 2).

Pace off the distance from your position 2 to the tree.

This involves far less walking and no math. Much better and quicker for uneducated soldiers in the haste and heat of the battle.

And for those who say a ‘Pace’ is horribly inaccurate I must point out that I’ve worked with soldiers who know their pace almost to the millimetre. They practice by marching a measured course and counting their paces over and over again at different cadences and speeds. Thus they know it takes X paces over smooth ground, Y paces over rough terrain etc. and can navigate in pitch black by dead reckoning to within a foot or so of their target over the course of a mile using only a compass and the counting of their paces.

answer says you must measure 90degrees. that is against rule that you cannot use measuring equipment (i accept you can use own body for foot length measure, despite original prhasing).

instead answer should be: when you reach tree, walk away from tree ensuring it remains directly in frount of river from your perspection, that has same effect without need for protracter.

Units for distance are a matter of agreement on a definition. Just define a new distance unit: the you-sign-distance, defined as the distance between you and the sign. Then, by definition, the sign is 1 you-sign-distance away.

Clever, but completely unnecessary.

Simply walk across the river, counting your paces to the sign.

Why do people insist on complicating things?

why do you insist on replying more than once?

Erm, read the puzzle again. You can’t cross the river.

You could jump in the river, float downstream and go to the local hardware store and buy a tape measure and acanoe and voila.Or go to the local bar and wonder why you need to measure the gap.

My solution is to use a cap.

I am wearing a cap deeply.

I Align the tip of the cap to the sign.

I will rotate toward the land without moving the head and neck.

I will find another object the tip of the cap.

I walk up to the object while counting steps.

Is this correct?

simple. find it on google maps on your smartphone.

i concur.

Very clever. I need to bone up on my congruent triangles. (:-)

my solution was much easier…just make a splash at your feet and the circular ripples in the river would flow out towards the opposite bank and to the right. when the ripples reach the opposite bank mark where the ripples are to your right.

You could look at the sun, estimate its angle from the top of its orbit, and, using an estimated height of the tree as a reference, use that estimate to estimate the length of the portion of the shadow it causes the tree to cast across the river (hopefully, the sun is behind you). Thus, river is ~x trees wide.

The solution with steps, though, is probably the most accurate. You’d probably know the length of your stride better than you could gauge the height of a tree.

Thanks Mark, glad you discovered it useful. :)