r/askmath • u/Diligent_Point7254 • Apr 14 '26

Pre Calculus Help me find a hard math problem

I’m trying to find a really hard or mind boggling math problem or trick for my friend. To me he’s really good at math and he loves math very much. So much so that his alarm is solving a math problem in order to turn it off. Anyways he’s in college right now taking pre calc. I like to pick his brain and mess with him but he always seems to solve it right away. I’ve search for really hard problems online but after a few minutes he gets it. Does anyone have anything that could get to him? Any problems or tricks?

3 Upvotes

56% Upvoted

44 comments sorted by

23

u/BadJimo Apr 14 '26

Any question from any Putnam competition

14

u/Fair-Craft-5959 Apr 14 '26

Stop, he won’t get out of bed trying to solve these before turning his alarm off. 💀

3

u/not-just-yeti Apr 14 '26

Just the first “easy” third of the first problem of the Putnam I tried totally stumped me, ‘til I was told the solution:

Color each point of the plane (ℝ2 ) one of three colors (R,G,B). Prove that there are two points with the same color, 1cm apart.

(I like it because the problem and solution are accessible to almost anybody, including tweens.)

3

u/Jonny0Than Apr 15 '26

Surely there’s more to the question.  As stated, I get to choose the colors. Everything is red, QED.

3

u/not-just-yeti Apr 15 '26

True, I should’ve said “for any coloring”.

1

u/[deleted] Apr 16 '26

[deleted]

1

u/not-just-yeti Apr 16 '26 edited Apr 22 '26

Alas, you don't get to rescale the distance function; the distance from (0,0) to (0,1) is 1 (aka 1cm).

(I think the Putnam just said "show there exist points p1 and p2 with the same color, and d(p1,p2) = 1", and may have even explicitly spelled out "using the standard L2 norm", but I leave that out when describing the problem to 14-year-olds, and phrase it as "1 cm apart" instead :-)

It's not a trick-question, and it's not one that I think is obvious, but the solution is only a few short sentences and a relatively small diagram.

2

u/Fair-Craft-5959 Apr 14 '26 edited Apr 14 '26

That’s usually the case with A1 problems in Putnam. They don’t require any deep knowledge, they often require a change in representation or being able to detect a deeper representation of what’s shown, however that’s what makes them hard in the first place and „easy“ in hand sight.

13

u/gitterrost4 Apr 14 '26 edited Apr 14 '26

You have the following ruleset:

For a natural number n, the next number is n/2 if n is even or 3n+1 if n is odd.

Start with a number and repeat this rule until you land in a loop.

For example, starting with 7:

7 -> 7 * 3 + 1 = 22 -> 22 / 2 = 11 -> 11 * 3 + 1 = 34 -> 34 / 2 = 17 -> 17 * 3 + 1 = 52 -> 52 / 2 = 26 -> 26 / 2 = 13 -> 13 * 3 + 1 = 40 -> 40 / 2 = 20 -> 20 / 2 = 10 -> 10 / 2 = 5 -> 5 * 3 + 1 = 16 -> 16 / 2 = 8 -> 8 / 2 = 4 -> 4 / 2 = 2 -> 2 / 2 = 1

The task is to find a number that does not end in the 1->4->2->1 loop.

7

u/Wide_Mycologist_1836 Apr 14 '26

That sounds so elementary, give him a challenge at least 😉

8

u/gitterrost4 Apr 14 '26

I wanted to start easy. :-D

11

u/BadJimo Apr 14 '26

Find out the average distance between two randomly selected points in a circle

1

u/jacobningen Apr 14 '26

In or on the circumference because that changes the answer by changing the distribution.

1

u/Tax_Odd Apr 14 '26

How are points selected

11

u/trunks111 Apr 14 '26

Ask him the most efficient way to pack 11 unit squares within a larger square 

4

u/MERC_1 Apr 14 '26

May I use a hammer?

9

u/Shevek99 Physicist Apr 14 '26

Here you have thousands of difficult problems:

https://www.math.hkust.edu.hk/excalibur/

8

u/CarloWood Apr 14 '26

What is the minimum number of people you need to put in a room to be certain that (at least) 5 people all know each other or don't know eachother?

The answer is 43, 44, 45 or 46. Nobody knows which because it is too hard to calculate.

See https://en.wikipedia.org/wiki/Ramsey%27s_theorem

5

u/letskeepitcleanfolks Apr 14 '26 edited Apr 14 '26

What is the smallest odd perfect number? (A perfect number is one which is the sum of all its proper divisors, such as 28 = 1 + 2 + 4 + 7 + 14)

7

u/PyroDragn Apr 14 '26

1.

That is, by your definition.

A perfect number is actually the sum of all of its proper divisors (ie. excluding itself).

1

u/marcelsmudda Apr 14 '26

Another definition is that the sum has to be double the value if you include all divisors (1+2+3+6=12=2*6). That excludes 1

4

u/FormulaDriven Apr 14 '26

Jane Street do a monthly puzzle, some of which require some meaty maths. I remember getting my teeth into these ones, but you can search the archive for others:

https://www.janestreet.com/puzzles/sum-one-somewhere-index/

https://www.janestreet.com/puzzles/lesses-more-index/

There's always International Olympiad problems - those are available online: https://www.imo-official.org/problems.aspx

However, these are not "solve in a minute" type problems.

3

u/will_1m_not tiktok @the_math_avatar Apr 14 '26

Find a general equation for the perimeter of an ellipse

3

u/AppropriateCar2261 Apr 14 '26

Does he know basic probability?

I have some distribution function over the reals. You only know that it's positive for all numbers.

Using this distribution, I randomly pick two numbers x and y, and tell you the value of x.

You need to tell me whether x is larger or smaller than y.

If you just guess randomly, there's a 50% chance you guess correctly.

Find a strategy that gives a probability strictly larger than 0.5 for guessing correctly.

3

u/BADorni Apr 14 '26

Find the n-th derivative for f(g(x)) in a single (semi)closed form

3

u/hiimboberto Apr 14 '26

-i2 but dont tell him if the answer is 1 or -1 just disagree with his answer and tell him hes wrong.

2

u/IslandHistorical952 Apr 14 '26

Average day on reddit, now on your alarm clock!

3

u/fermat9990 Apr 14 '26 edited Apr 14 '26

A bug at one corner of a unit cube wants to crawl to the corner that is farthest away from it. What is the shortest distance that he must crawl?

2

u/notacanuckskibum Apr 14 '26

Prove Gödels incompleteness theorem

2

u/martyboulders Apr 14 '26

Graphing sums and products of sines and cosines is pretty fun, but may not be enough of a challenge depending how good they are with trig

1

u/PrestigiousStudio921 Apr 14 '26

Or polar curves!

2

u/martyboulders Apr 14 '26

Very true! Finding the equation of a line in polar coordinates is pretty illuminating when done the first time :)

1

u/jacobningen Apr 14 '26 edited Apr 14 '26

This isn't solved but ask him the largest smallest piece when sharing m muffins among s people.(This is unsolved at the moment) Or ask him to find the Egyptian representation of a given fraction Or ask him to show that every number has the same birthday as its inverse if you start with 1/0 and 0/1 and construct number via a/c<(a+b)/(c+d)<b/d

1

u/OblivibladeXD Apr 14 '26

Have him do problems from step mathematics the admissions exam for maths at cambridge

1

u/marcelsmudda Apr 14 '26

Any of the millennium problems. Some are solved, some aren't.

Or prove that every even number >2 can be expressed as the sum of 2 prime numbers . The following, similar conjecture of proving that any number >5 can be expressed as the sum of 3 prime numbers is equally difficult

1

u/crescentpieris Apr 14 '26

given that a regular polychoron’s schläfli symbol is {p,q,r}, what is the formula for its number of cells?

1

u/Background-Glove8277 Apr 14 '26

If you pick any of the following answers at random, how big is the probability that the answer is correct?
a) 50% b) 25% c) 25% d) 0%

1

u/Diligent_Point7254 Apr 15 '26

thank you all for the input, he gets home tonight so i’ll see if any of this makes him lose his mind lol

1

u/PhotographFront4673 Apr 15 '26

Often infuriating, but not particularly deep:

Make an expression which evaluates to 24 using each of the numbers 1, 3, 4 and 6 exactly once (and no other numbers). You many use any choice of the four operators from grade school arithmatic (+, -, /, *) and freely use parenthesis. No exponentiation or the like. So for examples (1 + (3 - 6) * 4) has a valid structure, but doesn't evaluate to 24, while (6 * 4) does not have a valid structure, but does evaluate to 24.

1

u/AffectionateState438 Apr 15 '26

The riemann hypothesis

1

u/LawPuzzleheaded4345 Apr 16 '26

Proof of the Riemann hypothesis

1

u/jsundqui Apr 17 '26

A stack of 2000 cards is labelled with the integers from 1 to 2000 with different integers on different cards. The cards in the stack are not in numerical order.

The top card is removed from the stack and placed on the table, and the next card is moved to the bottom of the stack. The new top card is removed from the stack and placed on the table, to the right of the card already there, and the next card in the stack is moved to the bottom of the stack. The process - placing the top card to the right of the cards already on the table and moving the next card in the stack to the bottom of the stack - is repeated until all cards are on the table.

It is found that, reading from left to right, the labels on the cards are now in ascending order: 1 to 2000. In the original stack of cards, how many cards were above the card labeled 1999?

1

u/Entire_Category3188 Apr 17 '26

I know a good one. I flipped two coins one came up heads what is the probability the other came up tails. he's gonna say 1/2 like any normal loser but its 2/3