Very hard challenges

Here's where I'll be storing challenges that I believe are much too difficult for most students. These are puzzles where the chance of success is slim, either because the solution requires a tremendous amount of steps, or because the construction required is extremely creative.

Langley's problem

I prefer MindYourDecisions to Nick's Mathematical Puzzles for the presentation of both the problem and solution.

159. Eight odd squares
Nick's Mathematical Puzzles

Box-ladder puzzle

I doubt any student would come up with the clever trick presented, therefore, I'm not going to index this video for now.

A Classic - Ladder And Box Puzzle

The same question is also posed here.

The following problem requires many constructions, and lots of steps. This is way too challenging for most people, as you'll notice by reading the comments. Not even Presh solved this one!

Solve:  \(4^x + 6^x = 9^x\)

The prereqs for the following challenge are factoring quadratics by grouping, dividing one equation by another, and using u-substitution. This problem requires many steps. I believe this problem too difficult for the vast majority of students, so I won't be indexing it. Eventually I would like to offer a course for competition math and this problem may be part of that course.

Find \(x\) and \(y\) such that $$\begin{align} x,y \lt 0 \\[1em] x + y + \dfrac{x}{y} = \dfrac{1}{2} \\[1em] (x + y)\left(\dfrac{x}{y}\right) = \dfrac{-1}{2} \end{align}$$

What Is The Square's Area?

I will not be indexing this challenge because it requires more than one, in this case, two, creative constructions. I doubt the vast majority of students would think of these. In case I decide later to index this challenge, here are the prereqs:

I don't yet have the mathematical chops to determine how difficult this problem is. Therefore, it's sitting with the very hard challenges for now.

Looking through the comments, I notice this challenge was solved by a very small percentage of students.

The ladder-box problem

Here are at least two ways of solving it. The problem is likely too difficult for most students. Here it appears again, as an NRICH activity.

Ant on the surface of a cuboid

This problem is way too difficult.

Cutting rectangles into squares

This problem isn't too bad if you start by labelling the smaller squares. However, the way the video solves it is complicated, and is unlikely to be found by most students.

This challenge is likely much too difficult for the vast majority of students.

1998 IMO, Problem 6

Reading the comments of this video, I think it's a bit too hard for the vast majority of students. Most of the people who found the solution needed to draw the situation using Geogebra or something similar. Also, this problem was supposedly the last problem on the first round of a Belgium math olympiad, according to two YouTube commenters.

This problem is difficult. I doubt many students could find the answer. Not indexing it for now, but may later.

This challenge requires being able to factor differences of higher order powers, specifically \(a^6 - b^6,\) which I'm not currently covering. I may include this problem later.

This is way too difficult for most students. If this challenge goes anywhere, it goes in modular arithmetic.

This is way too difficult for most students. If this challenge goes anywhere, it goes somewhere in elementary number theory.

This puzzle would fit best with proving the divisibility tests using modular arithmetic, but I think it would be a bit too difficult for the vast majority of students. Here's its solution.

This is surely too difficult for students, as they've never seen a polynomial factored into trinomials.

Spokes by NRICH: This problem is way too difficult.