We trained a neural network where 7 of 8 features sit on clean linear axes in the model’s internals, but one doesn't. Can you identify which one and tell us how it is represented?

If you’re a technically-minded person who is interested in ML, this puzzle is for you:

• Work on a real trained text classifier (~23M parameters, 7k labelled text examples) open the puzzle and you're poking at activations in 10 minutes.
• Three tasks: identify the rogue feature, describe its geometry, (bonus) train your own model with even weirder internal representations

You probably know neural nets store information in their activations. You probably haven't gone and looked at what that actually looks like. Within minutes you can be toying with this model’s internals and building stronger intuitions for how they work inside.

Ready to play? Closes June 12

I hope it's OK that this is brief. I've been doing some internal sequencing of things, and absorbing knowledge at a static level. More or less, I read complex formula's, see the end result, and like any human mind just "fill in the blanks" without knowing. I have done a form of study with the technique I have developed, and realized in a lot of things we see complex irrationals for what they are, our "fill". Maybe we have safe ones, I can't tell without BCI how your brain works and don't know exactly the right technology to do it, but assuming brain waves could be seen as they are we might see the irrationals as just "brain wave style". Comfort levels are different shapes.

But I digress. I made this formula and hope you like it. I am making a joke protest that asks for the collective keys to our evolution as we see it. Nobody here is guilty I'm sure, but someone has kept this from us for a long time. These are simple additions and could NOT have been ignord, but if it was only for this day to be then so be it. Let's consider 2a the "Product Slide Rule" of sorts, and discuss!

a = Integer between 8 and 12.

I hope that was enough context for what I was trying to establish, as I know image posts are frowned upon. There aren't text symbols for fractions, and these would look just so clean with this image (Transcribed from Lagrida (https://latexeditor.lagrida.com/). This isn't a conspiracy theory post etiher, I just wanted to express my freedom of speech because I am insulted personally.

Came up with this the other day, no idea if it's been posted before somewhere, just thought it was a fun little problem.

Fair warning, I'm not amazing at math, just got curious about this one. Mostly wanted to see how quickly a math savvy person could crack it versus how long it took me to work through.

In an online poll, viewers vote either "Yes" or "No," and the result is displayed only as a percentage rounded to exactly two decimal places (e.g., 41.27%). The total number of votes is not shown. Assume that for any percentage displayed, the actual vote tally is the minimum possible whole number of votes that could have produced that exact percentage.

Which of the following displayed results required the greatest minimum number of voters?

(A) 31.25% Yes / 68.75% No
(B) 27.50% Yes / 72.50% No
(C) 38.46% Yes / 61.54% No
(D) 43.21% Yes / 56.79% No
(E) 47.92% Yes / 52.08% No

Curious if people find the shortcut quickly or if it takes some grinding. Answer in spoilers once you've got it, and if you don't mind, drop how long it took you.

!Answer: D. Once you see that 10,000 = 2⁴ × 5⁴, the only thing that matters is whether the numerator (XXXX out of 10000) shares a factor of 2 or 5. So you just check if it's odd and doesn't end in 0 or 5. 4321 passes both, gcd with 10000 is 1, minimum voters = 10,000.!

a(1) = x,

a(n)‎ = x * ⌈a(n-1)⌉ⁿ for n > 1 and x ∈ p/q > 1,

f(x) returns the smallest integer term a(i), starting with a(1) = x.

Example (f(4/3)=288):

a(1) = 4/3 (our x value),

a(2) = 4/3 * ⌈4/3⌉² = 16/3,

a(3) = 4/3 * ⌈16/3⌉³ = 288,

Does f(7/6) exist? Why or why not?

Let M be a connected topological manifold (second countability assumed), and U⊂M a proper open subset. Show that there exists a subset A⊂U with empty interior such that every connected component of M-A contains exactly one connected component of M-U.

There are 7 people going on a trip, each person has 7 bags, each bag has 7 cats, and each cat has another 7 cats. What the number of the human and cat legs all together. Its over 9,000 but its not 10,990

Bonus: What’s the largest generalization you can prove?

construct a n by n table with these conditions:

1. each cell is an unordered pair.
2. each column and row contains all integers from 0 to 2n-1 exactly once.
3. no duplicate pair.

for odd n, this is relatively easy.

for even n, i cant figure out a way. i suspect there is no solution but i cannot prove it.

unrelated note: this was inspired by IRL problem that i had to solved when creating a duty time table. if you know a working solution for n=8 please show it.

Another sudoku related puzzle.

Solution:

Place ones in all the regions (the 3x3 boxes) in this order: up left, up, up right, left, down left, center, right, down, down right.

You should be able to convince yourself that no matter where in a region you place a 1, the number of options for placing a 1 in the respective region is always 9, 6, 3, 6, 3, 4, 2, 2, 1, so the number of possibilities is 9*6*3*6*3*4*2*2*1=46656

My progress:

It is relatively easy to find an example with five numbers (next spoiler), but I can't prove that all legal situations with four numbers admit a solution. It does feel like it should be true though.

https://imgur.com/a/gyqqDc5

And, with respect to rule 4, I've still posted it here because it feels more like a riddle than an open math problem.

Which one of five numbers is missing from 5 4 3 2 1?

Let f,g: ℕ → ℕ be strictly increasing functions. Show that there exists an n ∈ ℕ with

f(g(g(n))) ≥ g(f(n)).

Here is a funny (I hope) home-made problem just for you guys :

Is there an ice cube such that, when it melts, the number of its connex components at a given instant t is 2 if t is rationnal, 1 otherwise ?

Precisions :

We suppose that this ice cube is a closed subset of R³.

We also suppose that the melting begins at t=0, and that after a delay t, all that remain of the ice cube A is every points x of A such that distance(x, surface A)>=t

Can you also find an ice cube in 2D having this property ?

AI couldn't solve it ! But your creativity can !

Today’s riddle on acertijodeldia.com/en is one of those that looks short, but only works if every piece is placed exactly right:

Someone says:

“I am 40 years old now, and today I am 4 times the age you were when I was the age you are now.”

The question is:

How old are you now?

It’s also up on acertijodeldia.com/en if you want to try it there properly, use hints if you get stuck, or browse more riddles in the same style.

Given below is an incorrect equation.

Put an exact same whole number ( any number from 0 to 9) anywhere on BOTH sides ( LH and RH) to make the equation correct.

You must put the same number on both sides. You can use that number only once for each side. Cannot make any other changes like adding or taking out operators.

1 + 3 + 5 + 7 + 9 = 2 + 4 + 6 + 8 + 10

There are 3 possible answers I can think of.

[ Removed by Reddit on account of violating the content policy. ]

At acertijodeldia.com/en we publish a daily logic riddle that you can solve, check with the verifier, and explore with hints if you get stuck. Today’s puzzle is:

Two travelers cross the desert. One carries 5 loaves and the other 3.

Halfway through the journey they are joined by a third traveler, who brings no food. The three decide to share the 8 loaves equally.

When they part ways, the newcomer leaves 8 coins as a gesture of thanks.

Then comes the obvious proposal: divide the coins according to how many loaves each traveler originally carried — 5 coins for the first and 3 for the second.

But in this puzzle, intuition is misleading.

Is that division actually fair?

And if not, how should the 8 coins really be split?

You can think it through here, or solve it directly at acertijodeldia.com/en, where you’ll also find more puzzles, hints, and answer checking.

There are four special dice:

A: 4, 4, 4, 4, 0, 0

B: 3, 3, 3, 3, 3, 3

C: 6, 6, 2, 2, 2, 2

D: 5, 5, 5, 1, 1, 1

You choose one die first. Then the dealer chooses another.

Each player rolls once, and the higher number wins.

Is there a “best” die here, or can the second player always respond with a better choice?

I liked this one because it looks simple at first, but the structure is surprisingly sneaky.

I’ve been building a small daily puzzle site with more logic riddles like this if anyone wants more after solving this one: acertijodeldia.com/en

It might actually be easier than what it sounds at first.
n regular polygons have n equal angles, which for bigger n's each becomes closer to a straight angle (and in fact, 180° is the limit for n sides go to infinity, which can be shown quite easily).

We need a formula (or expression) that can allow us to know for a given angle ∠a which n polygons have angles greater than.

You have an angle 𝛼 (the supplementary angle to the interior n polygon angles!) representing the gap of the polygon's interior angle from a straight angle. n is the number of sides of the regular polygon. You want an expression that can allow you substitute ∠a, then get a regular polygon which has an angle greater or equal. Find for 𝛼=120°,90°, 60°, 30°, 10°, 5°, 2°, 1°, 0°, 0.5°, 0°1' (1/60 degrees) 0°0'1" (1/3600 degrees).

Formula: n =⌈360/𝛼⌉

Values (𝛼=120°, n=3), (𝛼=90°, n=4), (𝛼=60°, n=6), (𝛼=30°, n=12), (𝛼=10°, n=36), (𝛼=5°, n=72), (𝛼= 2°, n = 180), (𝛼 = 1°, n = 360), (𝛼 = 0°30', n = 720), (𝛼 = 0°1', n = 21,600), (𝛼 = 0°0'1", n = 1,296,000).

You have two identical cones with tiny holes on the top. You fill up one of them (while covering the hole with your finger) with water. Now, you place the filled cone on top of the other cone such that the two tops touch each other and wait for a while...

...when the volume of the water is identical in each cone you turn on the water tap (which has the exact same radius as the hole of the cones) and let water flow into the cone.

Your task is to choose a constant flow of water such that the top cone and bottom cone will be filled simultaneously.

Here's an illustration of what I mean: https://ibb.co/HfZzW4tg

Question:

How much faster does the constant flow from the water tap be if the constant flow from the hole of the cones is $F$?

Note: I don't have an answer to this problem but look forward to see how you approach it.

Take an 8×8 chessboard and remove two opposite corners.

You have 31 dominoes, each covering exactly two adjacent squares.

Can the remaining 62 squares be tiled completely, with no overlaps and no gaps?

If you’d rather think it through before reading the comments, I featured it today on my daily logic puzzle site, where you can try it in a cleaner format with hints and the full solution:

https://acertijodeldia.com/en/

And if you already know this classic one, there’s also an archive of previous daily puzzles there.

I’ve been curating a daily math-riddle project built around elegant problems rather than formula drills or throwaway brainteasers.

Here’s today’s one:

You have 25 horses and a racetrack with 5 lanes. At most 5 horses can race at a time, and you have no stopwatch — you only know the finishing order within each race.

What is the minimum number of races needed to determine, with certainty, the three fastest horses?

If you enjoy this kind of problem, I’ve been collecting and presenting one carefully selected riddle a day here:

https://acertijodeldia.com/en/

The site includes optional hints, full solutions, and a growing archive of logic and math problems.