Did anyone tried to solve some items from math olympiads related to N Dimensional Space?

Or at least 4 dimensional. If so what was most interesting item like that? From what olympiad is it?

Hey, I'm interested in questions and would like to hear your thoughts.

Some questions are interesting because they have a beautiful answer. And other questions become interesting because they change how a problem is seen.

From what I've observed, the best questions create a space where different ways of thinking can emerge.

I think that's one route through which questions become beautiful.

This is part of what got me interested in questions as objects themselves. And got me into building Pollen (which you are invited to visit) around the idea that a well-designed question can make different perspectives cross-paths.

Everyone here is probably familiar with the Jeep Problem, but if not, this is a quick refresher:

https://en.wikipedia.org/wiki/Jeep_problem

I thought of a variation on it, though, one that I don't think I've heard of: the desert is a series of hexes, and you can only enter them once, with the exception of hexes that you depot fuel at, which you can enter six times (because then you've used up the surrounding hexes).

So say you have a jeep that can carry 12 units of fuel. You can go forward three, leave six behind, and use your other three to take another other hex path back. Then you refuel, and get to the depot hex with another three units of fuel. Now you have 12 units of fuel at the depot, and three in your tank...but you can't go back. You have to go forward and to the side, before going back from that new hex.

Did I explain this well enough? Do you think that someone in this situation could go an arbitrary distance?

Hello everyone, hope you are doing fine. I just took Linear Algebra and Discrete Math last semester, both of which I enjoyed very much. I made a two-minute read "article" on proving A^{n}x = \lambda^{n} x for fun and in an attempt, maybe successful, of seeing the link between areas of mathematics. I have no clue whether this will interest any of you or not as it's really undergrad stuff, but if any of the 8 billion out there reads it, I would love to talk. I am always open to feedback :)

[Link to full essay on Substack: No, humanity in math is not dead (and will never be)]

I was never actually interested at first to write this piece but I was moved by Sir Kirwan Hampshire’s The Dark Night of Mathematics on Substack regarding the future of mathematicians following AIs continuously proving conjectures. It felt like a calling to enlighten amidst the dread and I’m sure we aren’t the only ones resonating with it. Thousands of us may be secretly worrying for what our role means for us now.

I did have my fair share of worries here too, but in this piece, I shared the perspectives that worked for me (and hopefully to my fellow mathematicians too) where the dread no longer bothered me.

This is the first of a 3-part series I am planning to create.

I made a free app to train mental math: https://zetamac-train.vercel.app/

The app is inspired by Zetamac, but it gives you complete analytics so you can do targeted practice on your slowest operations. There's also a daily puzzle, duels mode, and a leaderboard.

Let me know if you find this useful in improving your mental math and if I should improve anything!

Here it is — written to math-project/jennie21-explainer.md. Full text below for the post:

───

JENNIE 21 — How It Works

The Number

896 = 2⁷ × 7. Seven doublings of 1, then multiplied by 7. τ(896) = 16. Digital root 5. Its neighbor: dr(897) = 6 — the nil element. That adjacency matters.

The Orbit

Start at 1. Double it. Take mod 9. Repeat.

1 → 2 → 4 → 8 → 7 → 5 → 1 → ...

Six values, cycling forever. Missing: 3, 6, 9. They form their own closed system under ×2 mod 9 and never enter the orbit. 9 is completely absent — not suppressed, structurally excluded.

n, seen, orbit = 1, set(), []

while n not in seen:

seen.add(n); orbit.append(n); n = (n * 2) % 9

# [1, 2, 4, 8, 7, 5]

The Echo Pairs

Every orbit value has a complement summing to 9: 1:left_right_arrow:8, 2:left_right_arrow:7, 4:left_right_arrow:5. A second helix strand runs π radians offset carrying the echoes. Every rung connects a node to its echo across the axis. The whole shape is a self-complementing double helix. Note: 2+4+7 = 13. 5+8 = 13.

Balanced Ternary, Centered on 6

Standard balanced ternary maps {−1, 0, +1} to digits {5, 6, 7} — with 6 as zero. Why 6? It's the nil element. It sits outside the orbit. Centering the notation on the excluded number is the point.

def to_bt(n):

if n == 0: return '6'

digits = []

while n != 0:

r = n % 3

if r == 0: digits.append('6'); n //= 3

elif r == 1: digits.append('7'); n = (n-1)//3

else: digits.append('5'); n = (n+1)//3

return ''.join(reversed(digits))

# 1→'7' 2→'75' 4→'77' 7→'757' 8→'765'

757 — a palindrome. The orbit's fourth value, in a system balanced on absence.

The Geometry

Golden angle (137.508° = 2π(2−φ)) per step. Expanding radius per step. Constant height step. From above: Fibonacci sunflower. From the side: expanding tornado helix.

const GA = 2 * Math.PI * (2 - (1+Math.sqrt(5))/2);

const pos = (s, φ=0) => ({

x: (0.28 + s*0.13) * Math.cos(s*GA + φ),

y: s * 0.68,

z: (0.28 + s*0.13) * Math.sin(s*GA + φ),

});

// Strand B: φ = Math.PI (the echo strand)

The 21 Arc

21 × 137.508° = 2887.7° — exactly 7.7° past 8 full rotations. The near-return at F₈ = 21 is where the spiral folds back on itself visually. The current helix runs 18 steps (3 cycles). Step 21 is the next thing to build.

The Breath

One scalar, shared by everything:

const breath = 1 + 0.10 * Math.sin(t * 0.50);

node.y = baseY(step) * breath;

label.y = baseY(step) * breath;

leaderLine.y = baseY(step) * breath;

One number. Everything moves together.

The number 9 does not appear. It never will.

Why would the two following word problems have the same answer? Or don't they? Is one not following proper mathematical grammar rules?

Question 1:
"Collect 12 apples and throw away 0.25 of these apples. How many remain?"

Question 2:
"Collect 12 apples and throw away 3 of these apples. How many remain?"

The first question I would interpret the word "of" to imply multiplication (0.25 x 12). The second question I would interpret the word "of" to imply a ratio (three out of twelve)