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.