How To Solve Oxford's Ladder Interview Question

Oxford ranks as one of the best universities in the world. Can you solve this ladder problem, which is adapted from an admission interview question?

Source

Desmos animation

Websites consulted in research

This is the only channel to feature math topics suggested by people around the world. Support the channel on Patreon so we can share the beauty of mathematics and make the world a better place:

If you buy from the links below I may receive a commission for sales. This has no effect on the price for you.

Show your support! Get a mug, a t-shirt, and more at Teespring, the official site for Mind Your Decisions merchandise:

My Books
"The Joy of Game Theory" shows how you can use math to out-think your competition. (rated 4.0/5 stars on 37 reviews)

"The Irrationality Illusion: How To Make Smart Decisions And Overcome Bias" is a handbook that explains the many ways we are biased about decision-making and offers techniques to make smart decisions. (rated 3.5/5 stars on 4 reviews)

"Math Puzzles Volume 1" features classic brain teasers and riddles with complete solutions for problems in counting, geometry, probability, and game theory. Volume 1 is rated 4.4/5 stars on 13 reviews.

"Math Puzzles Volume 2" is a sequel book with more great problems. (rated 4.5/5 stars on 6 reviews)

"Math Puzzles Volume 3" is the third in the series. (rated 4/5 stars on 6 reviews)

"40 Paradoxes in Logic, Probability, and Game Theory" contains thought-provoking and counter-intuitive results. (rated 4.4/5 stars on 13 reviews)

"The Best Mental Math Tricks" teaches how you can look like a math genius by solving problems in your head (rated 4.8/5 stars on 5 reviews)

"Multiply Numbers By Drawing Lines" This book is a reference guide for my video that has over 1 million views on a geometric method to multiply numbers. (rated 4.3/5 stars on 6 reviews)

Connect with me

[fraydizs]

Oxford interviewer: "answer this this question"
Me: "easy" *gives answer*
Interviewer: "correct... now prove it."
Me: *sweats profusely*

[Sourdoughgirl]

Easy. The answer is $50, 000. You see it works for the Hollywood parents.

[pilroberts]

Pythagorus as a young man, years before fame and fortune came his way, drafted his first theorem;

Don’t walk under ladders

[AjitSharma-kmev]

Amazing animation! I remember doing this problem in high school and it was amazing. But the fact that the case of falling away and falling against the wall have the same locus of mid-point is mind blowing. Physics always surprises me.

[jamirimaj]

Case 2 is literally the definition of a locus of a circle... so it's just weird "proving" a "definition"

[yuryorlovskiy]

You can also solve it with calculus.
slope of the point will be the same as the slope of the ladder.
We have dy/dx=x/(-y)
Solving the differential equation we get
x^2+y^2=C which is the equation of a circle

Correct me if I'm wrong

[eurovisioncyan]

You don't want to be on the ladder...

[Pining_for_the_fjords]

Son: "Dad, help me with my homework."
Dad: "I'll get my ladder."

[xenozodiac]

I was thinking in a different manner. The one where the ladder falls over is a quarter circle since the midpoint is always the same distance from the origin. In the case where the ladder slides down, the ladder orientation is the mirror position of the one where if falls over. Which means the midpoint is the same.
So, the path traced by the midpoint in a falling ladder is the same as the path traced by the midpoint in a sliding ladder.

[henryginn]

For all those who say it's too easy, if you say it's a quarter circle, you would do terribly. They want to see how you think and come to that conclusion, not just give some intuitive guess. They would also go on to give harder versions of the problem. They already know you are good at maths as you need to pass the MAT exam to get to interview anyway, here they want to test how you can break down a problem and see how you approach things

[JoeHuddleston]

I got this answer:

"I'm applying to Oxford to learn things like this. If I already knew it, I wouldn't need to be here."

[Nights]

I know why I won't be going to Oxford.

[rhydes_]

I have actually realised 2 things that sound self explanatory, but sometimes aren't:
1) there are, of course many solutions to 1 problem but what is inpressive about these riddles is the fact that solutions are TOTALLY different from each other. Thus, you don't HAVE to know something. Just choose another option.
2) all mathematicians have a "character", preferable ways of solving tasks etc. This person loves the Pythag. Theor. And you can realise that after watching about 5 videos of his.

[ThePracticalMechanic]

I completely understand what they are looking for and how to solve and prove the answer they want but having used a ladder many times in the past I know that the ladder legs normally have a 2-3”base and some have a pivoting base as well. This will change the arc depending what base is used at least when the ladder fall backwards off the wall. Normal use of a ladder also requires the base to place away from the wall to prevent the ladder from falling over backwards but if the ladder was to be setup properly and someone pulled the ladder off the wall you would end up with more than a quarter circle with the slight change in arc cause by the change in pivot point because of the wide base. Thanks for the video. It made me think a little harder this morning than I normally would.

[tunglai]

when I was in high school the question was to calculate the acceleration of that center of mass of the rod.

[alvinpalmgren]

2:25 If you slide the upper triangle down and turn it around (so that the two y-sides are facing eachother), an isosceles triangle is formed. In this triangle, the distance between the corner and the center of the ladder is equall to L/2. Since L/2 is constant, so is the distance between the corner and the center of the ladder. Hence the center of the ladder must follow the path of a quarter circle while falling (it's the only shape where the distance between the center and the circumference is constant)

[DuongNguyen-dedn]

Actually, you only need basic geometry to solve this.
Case 1: you have the line from the wall corner to the ladder's midpoint = L/2 ( prove this by drawing extra lines and the proof of 2 equal triangles ). Then have it as the radius(es) of the circle centered at the wall corner => you have a 1/4 circle cuz the wall corner is 90 degrees.
Case 2: even easier, you always have the distance of the center to the midpoint is L/2 so that easily form a 1/4 circle

[sapnamehta]

Simple
We can assume the point to be a variable point(h, k).Then we shall write using Pythagoras theorem .If the coordinates of the midpoint are (h, k) then the intercepts on axis are (2h, 0) and (0, 2k).
Now using Pythagoras theorem assuming length of rod to be 'l'..
We can easily write the locus of the midpoint as quadrant of radius l^2/4...

[powerofwisdom]

Easy! If the ladders length is L, and X is dislocation of the ladders buttom end, then the position of the ladders top end Y satisfies the expression: X^2+Y^2=L^2. Since the midpoint always has the coordinates X/2, Y/2, then for that poit we have (X/2)^2+(Y/2)^2=(L/2)^2
Which is the circle of radius L/2

[tweytwan]

For the first one, you could also say that the median from the right angle in a right triangle is equal to the length of the hypothenuse divided by 2, so this was a constant value for each case.
You can prove it because the intersection of the mediators (which is the center of the circumscribed circle of the triangle) is the middle of the hypothenuse.