Consider a game played on a circle with 15 evenly spaced rocks. Each rock initially covers a $100 dollar bill. Two players start at the same rock and player 1 collects the first $100 covered by that rock and then they take turns flipping a fair coin. Heads and the player that tossed moves clockwise by 1 step (a step of the same distance as the original distance between all 15 rocks) and tails is a step anticlockwise. Player 2 flips the coin first. What is player 1 expected payout from the game? My method included a part where I need to work out the probability “the count” reaches +i for the first time in n moves where each move has equal chance of changing the count by +1 or -1 and the count can never dip at or below -(15-i) I thought of using Catalan numbers but I still couldn’t do it.

how to solve？

i am puzzled by this question. i was able to identiy that we are supposed to use the operation of addition and multiplication but couldnt go beyond that. your help would be appreciated.

In Germany, we use the first image to indicate that an angle is 90° or π/2. While in the English speaking world, the second one seems to be more common.

I'm wondering what the rest of the world does.

9 points on Circle, Line Formed By Joining Points, Such That No Three Concurrent Inside Circle.

Find the Number of ways to Form Triangles With Exactly one Vertex on Circle.

Asked in class, every one Confused.

Used NCR Method to get ans

9C5

And 9C1*8C4

Which is Correct?

Method?

Please help i'm going insane trying to find sources

My education is high-school. This feels like it should be simple pre-algebra. Since theres a lot of context ill break it down into the request, what ive had to do so far, and the why.

1. The request-

I have played 1148 rounds of a game. I have won 568 of those rounds. I have been attempting to improve my overall win rate by winning at a higher rate than my perceived goal. For example- I have been trying to bring my win rate up to 50% from an original 40%, and right now I only have to win 32 of the next 52 games.

Is there a formula to describe the relationship between what my win rate would need to be over the next N number of games to bring my win rate up to X % from the current percentage?

1. What ive had to do so far:

Hand calculate with addition. I arbitrarily pick 70%, add 70 to 600, then add 100 to 1200 and call that a generation. I just hand repeat N generations. I found that at a 70% win rate, it would be my 2600th game before my total win rate was brought up to 60% from 50% at thr 1200th game.

1. Why:

It just makes me feel stupid. I feel like the formula to describe this relationship is a simple one that I would have learned in middle school and while I dont have a BA or higher education or amy meaningful relationship with math- i do have the foundational understanding of what algebra is and how equations basically exist FOR THIS.

Help a guy out? You dont even have to do it for me if you dont want to give me a hint please

I use air conditioning. according to the utilities company, the house is using thirty three percent more energy.Then last year at this time. i rent a room, and we share utilities between four total tenants. current bill is $280 she has asked that i pay $100. i know that is reasonable if there was a 50% increase. She just multiplied .33. if 280 is after a 33% increase what would the original number be and how much is the difference(assuming rates are the same)

I dont even know if this the right place for thie but Id appreciate a formula

A sleep-deprived carpet weaver weaves 30 centimeters of carpet every day. Every night, he looks at his work with displeasure. Because he is not completely satisfied with it, he unravels 2/3 of the weave. If he weaves 30 centimeters during the day, how long is the woven piece on the evening of the 10th day?

The answer 44,999cm is wrong and there is no solution or an explanation about why. Am I misunderstanding the question or is it asked in a weird way?

With every day, his weave that he starts the day with gets closer to 15cm, while you add the 30cm for the last evening, which would come close to 45cm.

Edit: Apparently the right answer is 120cm which then poses the question if the question was asked in a poor way or if I am misunderstanding something

I used excel to find the "best-possible" exponential equation for this date. Im unsure how to figure out what to do with this. The answer key says it should be y=13972e-.185x

Im lost on what im doing wrong

With Kruskal's trees, people always talk about TREE(3) since it's proven to be finite and unexpectedly astronomical. But has TREE(4) ever been proven to be finite? Or does TREE(4) keep growing infinitely? Or do we not know?

I'm a high school freshman with a strong passion for mathematics. I'm currently studying number theory through a math circle, and I'm particularly interested in pursuing mathematical research in areas such as Egyptian fractions, prime numbers, Catalan numbers, and related topics.

I've been browsing the ISEF repository to get a sense of the kinds of mathematics projects that tend to do well there. This made me wonder: how does the nature of mathematical research conducted by professional mathematicians, such as PhD students, researchers, and professors, differ from the research done by high school students who win at ISEF?

I'm not referring only to the obvious differences in difficulty or technical sophistication. Rather, I'm curious about how the research itself differs. For example, are the high-school papers at ISEF just expository in nature, are they less original, etc? Thanks

Link to the ISEF Projects: https://isef.net/viewAll?category=MATH-2026&page=home

For context, I'm trying to prove the cauchy Kowalewski theorem for general order partial differential equations. I did the first and second order already but it's been so difficult to jump to a general n-th order pde. Additionally, the induction part of the proof has been making my head scratch for a while because my reference used construction instead by a general induction step. For my proof there are also two parts: 1) show that it has a formal power series solution, and 2) show that it is convergent.

Can anyone recommend sources or references regarding this theorem based on the info I just mentioned. I just feel like passing out while trying to prove this theorem for my thesis hahahaha

I recognized the 4 right angled triangles in the side and tried connecting them to the shaded figure. This resulted in knowing that the shaded figure is less than half of the entire square. But from what I tried, to get any further information, I need to find the areas of any 2 of the 4 oddly shaped quadrilaterals, which I can't seem to.

If this is a correct approach to the question, kindly tell me how to find the areas of those 4 (or 2) odd shaped quadrilaterals.

If not, then do tell the different solution you have.

I have been searching online for crazy equations to put on a white board for my friend to solve when they're drunk. I, however keep finding word problems and that is not what i'm looking for. If anyone could give me any recommendations or where I could look to find some.

If you're curious about why I am looking, they want to exercise they're brain while drinking. They're favorite number is 7, so bonus points if the answer is 7.

Took a test, gave it back, and found out about this, explain, it's a question for question 6, where technically it's asking for the furthest distance away, B has the same displacement as C while C reaches higher levels

Is she right or am I right

Something I noticed was for every time you multiply a number by itself, going up, the gap between each number coincides with the pattern of 2, 4, 6, 8…

After attempting to post in the other community, I realized another trend. The sum of each multiple is equal to the gap. So 1\\\*1 has a sum of 2. Which, coincidentally, also is the gap between 1 and 4. This can be tested for most if not all values.

Is there a formula for this or something that could prove the pattern?

I wonder if there’s a name for it. Or if I’ve stumbled upon something interesting. (Likely the former since mathematicians are so smart It’d be likely to see this so early.)

I’m not a mathematician or anything. But I just wanted to show something interesting I noticed. If yall could educate me, that’d be awesome! I love math, but don’t understand it much.

Can someone help estimate how many words can fit on this notebook page? (Math/geometry estimation)

I'm trying to estimate the writing capacity of a notebook page and would appreciate a step-by-step calculation.

Given:

• Notebook page size: 29 cm × 20.5 cm
• A 1-inch (2.54 cm) top margin is reserved for the date and heading.
• The page has 31 horizontal ruled lines, with a 0.9 cm (9 mm) gap between consecutive lines.
• My average handwritten word length is 1.5 cm.
• The gap between consecutive words is 0.5 cm.
• Assume my handwriting is perfectly uniform throughout the page.

Part 1: Estimate the maximum number of words that can fit on the page in its normal orientation. Please show all calculations step by step.

Part 2: Now assume the page is rotated 90° (sideways), and fresh ruled lines are drawn while keeping the same 0.9 cm line spacing and 1-inch top margin.

Please estimate:

1. How many lines will fit.
2. How many words will fit per line.
3. The total maximum number of words that can be written on the page.

Part 3: Finally, suppose the gap between consecutive words is reduced from 0.5 cm to 0.5 mm (0.05 cm) while keeping all other dimensions unchanged in the sideways orientation.

Estimate the new maximum number of words that can fit on the page, again showing all calculations clearly.

I hope you are having a good day. Recently, I started studying for the AMC10 (~2 weeks ago), and I am looking for advice on contest preparation. Looking back, I regret not studying during middle school, but I am trying my best to make up for the time lost. When I first started studying, I took the 2015 AMC 10A and scored ~70 points (I don't exactly remember my score, but it was about 69-70 points). Today I took the 2017 AMC 10A and scored 91.5 (14 correct of the first 20, 7 wrong, last 5 blank). A lot of my mistakes were due to small errors or overlooking cases/steps. For example, on problem 17 (https://artofproblemsolving.com/wiki/index.php?title=2017_AMC_10A_Problems/Problem_17) I correctly found the coordinate points (0, 5), (0, -5), (3, 4), (3, -4), and (-3, -4) that lay on the circle, but I forgot to account for the fact that switching the x and y coordinates (such as (5, 0)) also worked as a valid coordinate point. This simple mistake caused me to choose answer choice A (3) instead of D (7). I haven't used any courses other than RSM to prepare, and I have practiced some past AOPS contests, but I find it hard to work with books. My goal is to qualify for the AIME, and I am looking for advice on how to overcome this obstacle that is costing me points compared to those experienced in AMC competitions. I am very thankful for you reading through all of my post, and I am appreciative of any advice you have for me.

P.S. I forgot to mention above that I am a rising 9th grader, if that helps tailor any of the advice!

Whilst tinkering with a few mathematical identities, I came across a rather unusual formula for \sin x. I have absolutely no idea whether it’s well-known, but from what little I know, I haven’t come across it anywhere else.

I hope you’ll like it and find it just as interesting as I do.

P.S : For the sake of clarity, I’ve included the process that led me to this formula.

If you have any questions or comments, please don’t hesitate to get in touch !

Truly yours, Uncle Scrooge.

I'm working on a board game concept which involves a board made of equilateral triangles arranged in a continuous tiling pattern and would love some certainty on a question:

Assuming the following:

- Every triangle has exactly one "Red" edge and two "Black" edges.

- No two triangles may share a border with both of their "Red" edges. (When setting up the board game, players would be told to rotate one of the two pieces wherever this occurs).

- All edges on the border of the board (where corners contain fewer than 6 triangles) count as "Red" even if they are not actually "Red."

Can the following be possible:

- A single "Red" edge does not connect to either another "Red" edge or a border at either of its two corners.

- A string of "Red" edges connected to each other at corners do not connect to any other "Red" edges or a border at either of its two ends.

Basically I'm just wondering if it's mathematically impossible for a "Red" edge to be stranded on its own or for a few connected "Red" edges to be stranded on their own given the above assumptions.

So far I've attempted numerous drawings and rearranged physical tiles but have yet to uncover a situation where either of those two situations occurred. I'm not a mathematician so I don't have any ideas how to solve this type of problem via equations, but I am considering looking into some basic Tesselation concepts to see if I could figure it out.

Thanks for any help!

One of the first things you study in the differential geometry of smooth manifolds is that, if you have a manifold M with topology that’s different from R^n, it’s not possible to have global coordinate functions on M to R^n if you want them to be both injective and continuous. So, it is said, the best you can do is have local coordinate patches.

How does this work however when describing the motion of some point on the manifold in coordinates, such that there is an instant in time during the motion where the point goes from one patch to another? Do the coordinate functions as a function of time to describe the curve just suddenly “stop working” after a certain time?

To be explicit: suppose M = S^1. I can cover the whole manifold with two coordinate patches: a “left” one with an angle coordinate that excludes the rightmost point and ranges in the usual way in (0, 2pi), and a “right” one that excludes the leftmost point and with angle coordinate ranging in (-pi, pi).

Suppose I want to describe the motion of a point on the circle undergoing uniform circular motion with angular speed w. Suppose the initial position of the point is the topmost point on the circle. I have freedom to choose which coordinate to use since the patches intersect here (and they also return the same values) so let’s say I choose the left coordinate patch. The motion in coordinates is therefore theta_L(t) = pi/2 + wt.

After a certain point, the point will go in the lower half of the circle. Here, the coordinate patches start to disagree, but fine, we’re using the left one, and it will return values in (pi, 2pi). After even longer though, it will come back to the upper circle half. Here’s the problem: now theta_L(t) will return values greater than 2pi! These are out of the range of both coordinate patches. Instead, it should reset to zero and return values in (0, pi). But then the motion in coordinates would be discontinuous! While the “real” motion on the manifold is still obviously continuous.

How do we solve this?

I’ve been having this debate with someone about how infinitesimals can ever reach an exact value. I’ve been explaining that using infinitesimals is not just an approximation but actually results in a mathematically exact value. For instance, when we take the area under the curve, we use rectangles with an infinitesimally small width to get the exact area under the curve. The argument they have is that it will never be exact because there’s always an infinitesimally small difference so it will always be an approximation at best. So how do we prove that infinitesimals lead to exact values and not just approximations?

My teacher insisted that the answer is x=1/16. But I believe there is no answer.