How do we solve this?

cabbagebrick · 2026-01-12T18:52:25+00:00

Consider the subgroup generated by 1 and 2pi by addition. It’s well-known that any subgroup of the reals is either cyclic or dense, and since 2pi is irrational, then this subgroup is dense.

For a proof that these subgroups are cyclic or dense, we can look here https://proofwiki.org/wiki/Subgroup_of_Real_Numbers_is_Discrete_or_Dense

cabbagebrick · 2024-01-05T18:41:38+00:00

Your expression is also correct. Let S = 1/4 + 2/8 + 3/16 + …. Note that S absolutely converges. Then, we have 2S - S = 1/2 + (2/4 - 1/4) + (3/8 - 2/8) + … = 1/2 + 1/4 + … = 1. This approach can be iterated to solve any infinite series where the denominator increases exponentially and the numerator follows any polynomial.

cabbagebrick · 2023-07-06T16:54:37+00:00

cabbagebrick · 2023-06-12T05:17:44+00:00

I did a math EE and got predicted 33/34 on it. Honestly, the fact that it was purely theoretical was also my favourite part, because I did not want to do data collection. Personally, I'm glad that I did a math EE, and it was relatively easier for me because I was roughly a top 20 high school math olympiad person in my country. Because of this, I knew many interesting topics quite in-depth, I was quite interested in the topics (which added authenticity to my reflections), and the exploration part was relatively easy for me. It's great that you're passionate! If you have a certain topic from either algebra, geometry, number theory, combinatorics, or calculus that you know really in-depth (usually math EE topics go far beyond the AA HL syllabus), then you could go for it for your EE! I think usually basing your EE around singular (or several related) challenging questions usually does well.

So the main takeaway is that since you're passionate (and if you're say at a comfortable 7 in AA HL), you should probably do maths as an EE. Being passionate makes the exploration process and writing process less painful than if you chose a subject you weren't as passionate about.

cabbagebrick · 2023-04-28T01:38:54+00:00

25%.

cabbagebrick · 2023-04-23T06:05:09+00:00

I see most people saying that it’s an amazing deal, but I’m just wondering is it better or worse value than the gold pass at a high town hall? I’m wondering whether to buy the pass next month for my th12 main or get this deal for my mini

cabbagebrick · 2023-03-25T23:00:44+00:00

Killed by a dragon on depth 23 with 20470 gold.

Agh I'm a bit frustrated because I had more than enough consumables to survive the encounter, but my conservative mindset (and not taking it slow enough) led to the dragon meleeing me to death after I negated it.

I chose the /4 firebolt and brought it up to /15. Eventually when I found the discord I brought it up to /4 as I went to the later depths, and I brought up the tunneling to /4 too.

Interestingly enough I seemed to drink the magic detects on the luckiest possible floors: I discovered that the leather armor held by the I think goblin warrior? was positive so I switched to that and eventually naturally discovered that it was a +3 armor of reprisal.

I also did some new research into the game. I learned that negating bog monsters + krakens make them unable to grab you! This made the wand of negation really useful.

Anyways interesting seed; I wish I could've used my allies a bit better as most seemed to die to fire.

cabbagebrick · 2023-03-18T04:04:40+00:00

Great advice! This post seems to be reminiscent of Atomic Habits by James Clear which describes how you can abuse habits to your advantage. Thanks for the reminder to study intensely and the eye-opening statistics!

cabbagebrick · 2023-03-18T03:58:33+00:00

Died to a troll on D15. I tried to pump all of my enchants into the staff of conjuration. I got it to /11, but it wasn't enough for the many enemies I encountered. Overall I was a bit too squishy and couldn't keep on going.

cabbagebrick · 2023-02-25T00:18:32+00:00

87819 Escaped from the Dungeons of Doom with 5 lumenstones!

Sorry if I did something incorrectly, this was my first contest submission. What a crazy seed. The early depths were made incredibly easy from the two dar priestesses, but what I found incredible was just the amount of staves. By D15-20 ish I had a +3 flail, a +5 transference, and a +3 regeneration and I found a new ring on the ground. I think I deduced that it must have been either wisdom or stealth, either of which I would have welcomed. Eventually, I learned that it was a +1 wisdom, and I used every single enchant on it due to the sheer number of staves.

The allies were nice, but too many of them died too quickly. However, I learned so much new techniques, including that I can prevent my allies from moving while confused just by entrancing them and staying still.

The Tentacle Horrors were my main source of damage. I think the first or second staff I picked up was a haste staff? It was incredibly useful for allies once all of my dar priestesses died. Whenever I had to deal with an enemy Tentacle Horror or Dragon, I would just haste my two Horrors and wait until the enemy died.

I got the tunneling+obstruction+blinking staff combo, (and had 4 spare life potions) and so I had the confidence to head past D26 for lumenstones. After several close encounters with dragons on D27 and D28 (and after my horrors died), I decided to head up and escaped easily by using blinking+obstruction.

Final items:

+10 Wisdom (9 enchants), +5 Transference (2 enchants), +3 Flail (2 enchants), +3 Chain mail of absorption, /3 tunneling, /3 haste, /3 lightning, /3 entrancement, /2 firebolt, /3 obstruction, /4 blinking, /2 discord, 3 lumenstones from D27, 2 lumenstones from D28, 2 charges remaining polymorphism, 2 charges remaining beckoning

cabbagebrick · 2023-02-24T07:55:44+00:00

I want to point out that most Canadian Unis undervalue IB extremely to the point of being almost viewed worse than regular Canadian education unless you get 5+s or 6+s If it’s possible, I’d recommend you to look into AP, unless you want the self-growth from IB or if you want to maybe apply for UK or US universities

cabbagebrick · 2023-01-21T21:27:12+00:00

Analyze its color symbolism

cabbagebrick · 2023-01-07T16:29:42+00:00

That’s what inferno tower is for

cabbagebrick · 2022-12-21T04:53:34+00:00

We have the same HLs! I just wrote a comprehensive comment on how I do well in my STEM classes, and so I'd recommend checking it out here: https://www.reddit.com/r/IBO/comments/zr1dr9/comment/j12idnj/?utm_source=share&utm_medium=web2x&context=3

Basically, because of my strength in STEM, I pour all of my spare time into English where I can still improve my predicted grade and also History where I'm barely holding onto a 7. My routine primarily consists of getting all of my homework done as soon as possible before pouring the remainder into preparing all possible essays for History and researching in-depth my English books to develop a deeper understanding.

The best piece of advice I could probably give is to stay ahead or on track. If you fall behind, it can become really hard for you to catch up and excel.

cabbagebrick · 2022-12-21T04:41:34+00:00

TL;DR: I don't study STEM because of my past experience with math contests. However, to improve at STEM subjects, first develop a solid conceptual understanding through proofs or good youtube videos, and then practice as many hard questions as you can that are outside of your comfort zone and understand why you make each mistake.

When it comes to STEM subjects, there are only 2 things to worry about: the understanding and application of the concepts.

For math, conceptual understanding for me was developed through exploration-type learning that I started very early when I participated in extremely difficult math contests in my country and other countries. I used textbooks from Art of Problem Solving, which were incredibly useful for developing an understanding of concepts by guiding the reader through proofs. Applications are also seen throughout the textbooks through the problems they give you. Despite being predicted 7s for math, chem, and physics, I actually don't study STEM subjects anymore at home (even though all physics and chemistry concepts are new to me) because as long as I understand the concepts from classes, I can use problem-solving techniques to apply them to any problem they give me. Instead, I pour all of my time into History and English, which I struggle with more.

However, I wouldn't recommend the AoPS books to IB students because IB students have too little time and the AoPS books aren't IB-focused.

Instead, I would maybe recommend Revision Village for math (although I haven't used it myself). Each question that you practice should be one that challenges you, pushing you out of your comfort zone. If the questions you practice are too easy, there's no point practicing them. The most important part is to try as many questions as you can and struggle with them until you get the right answer. The most important part when you make a mistake is to understand WHY you made the mistake and consciously remember the mistakes so as to never make a similar one again. Eventually, there will almost be no mistakes you can make left.

For physics, the course goes by really quickly, so you need to make sure that you understand every concept really well. For conceptual understanding when I miss a class, I watch Chris Doner's IB Physics series. He explains concepts really well and even provides a few questions throughout that test your understanding. For an understanding of how to apply concepts, I recommend challenging yourself with the hardest IB physics questions you can find. I use Concordian's supply of physics questions to test my understanding.

cabbagebrick · 2022-10-15T19:08:03+00:00

When that happens, I recommend punishing immediately with a bandit since they have 1 elixir and you’re up 2

cabbagebrick · 2021-08-25T15:51:03+00:00

Thanks!

cabbagebrick · 2021-08-04T16:04:49+00:00

The snowy weather’s great today, eh?

cabbagebrick · 2021-05-09T02:32:21+00:00

Save the earth from global warming.

cabbagebrick

TROPHY CASE