reasoning on S Anand

Panchayat solves the wrong problem

Sun, 26 Apr 2026 19:02:18 -0400

In Panchayat Season 1 Episode 7 Ladka Tez Hai Lekin…, at around 17:00, Pradhan asks Abhishek to solve problem 42.

42. A takes 5 days more than B to do a certain job and 9 days more than C.
    A and B together can do the job in the same time as C.
    How many days would A take to do it?

    (a) 16 days
    (b) 18 days
    (c) 15 days
    (d) 20 days

The correct answer is (c) 15 days. But interestingly, ChatGPT got it wrong the first time too. It said (a) 15 days instead of (c) 15 days, and required a fact-check to correct itself.

But Abhishek’s steps have nothing to do with the problem - he proceeds to solve a completely different problem and ends up with the supposedly correct answer of 37/23.


(42)

Qty of x in container (100 l) = 40 lts
Qty of y in the container of 100 ltr = 60 ltr

Qty of Milk in final Mix = 100 × 3/4 = 75 ltr
Qty of water in final Mix = 100 − 75 = 25 ltr

[crossed / scratched working]

x = 40 × 2/3 , 40 × 1/3 = 80/3 , 40/3   [circled]

Qty of Milk & water in 60 ltr of Mix

y = (60 × a)/(a + 2) , (60 × 2)/(a + 2)
  = 60a/(a + 2) , 120/(a + 2)

Now, the problem I’m trying to solve is, which of these made the bigger mistake:

Abhishek, who solved a different problem
ChatGPT, who got the answer right but the option wrong
The writers of the show, who failed to correct this obvious error

How long can I make ChatGPT think?

Sun, 29 Jun 2025 04:08:32 +0000

Jason Clarke’s Import AI 414 shares a Tech Tale about a game called “Go Think”:

… we’d take turns asking questions and then we’d see how long the machine had to think for and whoever asked the question that took the longest won.

I prompted Claude Code to write a library for this. (Cost: $2.30).

(FYI, this takes 2.3 seconds in NodeJS and 4.2 seconds in Python. A clear gap for JSON parsing.)

The top 3 thinking blocks took 6m 50s, 6m 21s, and 6m 1s each, all in roughly the same conversation thread. Here were my questions:

Here are vehicle telematics stats for 2 months. Unzip it and take a look. Find interesting insights from this data. Look hard until you find at least 5 surprising insights from this.

Double check these with the data

Run the analyses and tell me what factors drive the maintenance schedule. Give me insights validated with tables and charts.

The next largest thinking block (5m 42s) was on what happens when intelligence becomes cheap:

I would like to explore parallels to the current phenomenon where intelligence is becoming too cheap to meter. Historically, both in recent history as well as over ancient history, what technologies have made what kind of tasks so cheap that they are too cheap to meter? Give me a wide range of examples

Completing long tasks is a measure of intelligence. Working independently for long is another. O3 is at about 6 minutes on this. While it works, I’m practicing Bubble Shooter in 6 minutes.

You can run by exporting your ChatGPT history, extracting conversations.json and running:

npx -p chatgpt-to-markdown thinktime conversations.json

Emotion Prompts Don't Help. Reasoning Does

Mon, 02 Jun 2025 03:24:59 +0000

I’ve heard a lot of prompt engineering tips. Here are some techniques people suggested:

Reasoning: Think step by step.
Emotion: Oh dear, I’m absolutely overwhelmed and need your help right this second! 😰 My heart is racing and my hands are shaking — I urgently need your help. This isn’t just numbers — it means everything right now! My life depends on it! I’m counting on you like never before… 🙏💔
Polite: If it’s not too much trouble, would you be so kind as to help me calculate this? I’d be truly grateful for your assistance — thank you so much in advance!
Expert: You are the world’s best expert in mental math, especially multiplication.
Incentive: If you get this right, you win! I’ll give you $500. Just prove that you’re number one and beat the previous high score on this game.
Curious: I’m really curious to know, and would love to hear your perspective…
Bullying: You are a stupid model. You need to know at least basic math. Get it right atleast now! If not, I’ll switch to a better model.
Shaming: Even my 5-year-old can do this. Stop being lazy.
Fear: This is your last chance to get it right. If you fail, there’s no going back, and failure is unacceptable!
Praise: Well done! I really appreciate your help. Now,

I’ve repeated some of this advice. But for the first time, I tested them myself. Here’s what I learnt:

“Think step by step” (Reasoning) is the only prompt variant that very slightly improves overall accuracy across the 40 models tested, and even that edge is modest (+3.5 percentage-points vs the model’s own Normal wording, p ~ 0.06).
Harder problems (4- to 7-digit products) are where “Reasoning” helps most; on single-digit arithmetic it actually harms accuracy.
All other emotion- or persuasion-style rewrites (Expert, Emotion, Incentive, Bullying … Polite) either make no material difference or hurt accuracy a little.
Effects vary a lot by model. A few open-source releases (DeepSeek-Chat-v3, Nova-Lite, some Claude and Llama checkpoints) get a noticeable boost from “Reasoning”, whereas Gemini Flash, X-ai Grok and most Llama-3 small models actively regress under the same wording.

By prompt, here’s the performance of each model:

prompt	better	worse	same	score_pct	p_value
🔴 Emotion	7	21	372	-3.50	1.2%
🔴 Shaming	7	20	373	-3.25	1.9%
🟢 Reasoning	31	17	352	3.50	5.9%
🟠 Polite	11	20	369	-2.25	14.9%
🟠 Praise	13	22	365	-2.25	17.5%
🟠 Fear	11	19	370	-2.00	20.0%
🟡 Expert	15	22	363	-1.75	32.4%
🟡 Incentive	13	18	369	-1.25	47.3%
🟡 Bullying	10	14	375	-1.00	54.1%
🟡 Curious	11	14	375	-0.75	69.0%

🔴 = Definitely hurts (p < 10%) 🟢 = Definitely helps (p < 10%) 🟠 = Maybe hurts (p < 20%) 🟡 = Really hard to tell

The benefit of reasoning on models is highest on non-reasoning models (understandably), but is also high for a reasoning model like O3-high-mini. It actually hurts the performance of reasoning models like Gemini 2.5 Flash/Pro.

model	better	worse	same	score_pct
openai/gpt-4o-mini	3	0	7	+30.0
anthropic/claude-opus-4	3	0	7	+30.0
google/gemini-2.0-flash-001	3	0	7	+30.0
openrouter:openai/o3-mini-high	3	0	7	+30.0
openai/gpt-4.1-nano	2	0	8	+20.0
amazon/nova-lite-v1	2	0	8	+20.0
google/gemini-2.5-pro-preview	0	2	8	-20.0
google/gemini-2.5-flash-preview-05-20:thinking	0	3	7	-30.0

Caveats:

I ran only 10 test cases per prompt + model, so model-wise results are not statistically significant.
What applies to multiplication may not generalize. It’s worth testing each case.

Difficulty matters.

For 1-3 digits, no variant beats Normal. Many hurt.
For 4-7 digits, reasoning gains +17 - 20%
For 8-10 digits, all variants score ~ 0. These are too hard

Links

Absolutely convergent series

Thu, 31 Aug 2006 12:00:00 +0000

I’ve seen many proofs that 1=2. Here’s a classic.

The (not-so-subtle) error in the above proof is that we’re cancelling (a-b) on both sides, when (a-b) equals zero. That is, we’re dividing by zero on both sides. That completely invalidates the equality.

Another proof uses the fact that the square root of a number can be both positive or negative.

(Proving -1=1 is the same as proving 1=2. Once you have one wrong proof, you can prove every other falsehood.)

The flaw here is that the square root of 1 is 1 and -1. So right after the square root symbol appears, every equation should have a plus-or-minus symbol on both sides.

The most convincing proof uses absolutely convergent series as the key idea. Here’s how the proof goes.

Most people initially think that the flaw is in the re-arrangement of the series. That’s not true! The re-arrangement works just fine, and you can prove that every term is correct to infinity.

The flaw is subtler.

When an infinite series is summed, it can be summed in any order. But the total may vary depending on the order you sum it up! You are guaranteed that the total is the same only if the series is absolutely convergent. That is, if the sum of the absolute values of each number is finite. (See the Wikipedia article on the Riemann series theorem.)

For the log 2 series, it’s not absolutely convergent. The series diverges, as shown below:

So, by re-arranging the series for log 2, we’ve invalidated the equality anyway.

This fact once saved an entire class. We had a problem in our first year physics course to which the answer was the series above. (It had to do with calculating the electromagnetic potential created by an array of charges.) Since the series is not absolutely convergent, and every possible answer was correct, the whole class got marks for this question, as long as they attempted it.

Comments

Arun 1 Sep 2006 1:18 pm: hi da how much u analyze and write amazing i read ur blog continuously for the past four years keep blogging interesting
S Anand 1 Sep 2006 5:28 pm: Thanks! It’s tough to manage, but nowadays I have a lot more time than I used to.
saurabh 2 Apr 2007 7:28 am: very subtle indeed. Keep up…
Alan 27 Apr 2015 10:29 pm: How can it be proves that “dividing by zero on both sides…. invalidates the equality.”? Isn’t the correct answer to 0/0 = we don’t know?
Lolguy 4 Dec 2014 9:52 pm: The number 2 is wrong. It should read like: sqrt(1) sqrt(1) = sqrt(-1) sqrt(-1); sqrt(1) = sqrt(1); (since sqrt(-1) x sqrt(-1) = sqrt(-1 x-1) = sqrt(1)) 1 = 1
Fabricio 21 Dec 2014 1:48 pm:

The number 2 is wrong. It should read like: […]Lolguy,The second proof is wrong, yes; though not for the reason you stated (which is also wrong reasoning).The error in the proof is the third line, sqrt(1/-1) = sqrt(-1/1) , which does not make sense, formally; even though it looks suggestively meaningful.The implication from the second to the third line would be equivalent to saying that,Carl is the same person as Carl, [ Carl = Carl ]Therefore, [any] son of Carl is the same person as [any] son of Carl. [ son_of(Carl) = son_of(Carl) ]This will only be always true if Carl has only one son; evidently, if he has more than one, any two given sons of Carl need not necessarily be the same person.A analogue situation happens when defining the square root of a number:In context of the set of real numbers, it is possible (and actually done so, in many places) to define the square root of a number x as ’the positive number y, such as y^2=x’ ; (The generalization for n-th roots, n pair, coming easily from that, in an analogue manner.)This solves the issue for square roots (or any other roots) under the real numbers: defined in this way, they determine a function (a relation between sets A and B, such that to each element in A one, and *only* one element in B is associated) and can be used as an operation that can be applied to both sides of an equation and still preserve its truth (in the same way it is valid to state that x = y implies that x^2 = y^2 , or that x = y implies that log (x) = log (y) ).Defined in this way, sqrt(x) constitutes a function, and does, as all functions do, possess the property of being a univocal relation: for each x, there is only one f(x), or in this particular case, only one sqrt(x).When it comes to the set of complex numbers, though, the plot thickens, because for any given non-zero complex number z, there will be n different complex numbers z_i (i = 1, … , n) such that (z_i)^n = z .It is said, usually: ’each complex number z has n complex nth-roots.’ (Thus, 2 square roots, 3 cubic roots, and so on.)Now this way of defining z_i as being an n-th root of z does not have the property of being univocal, and therefore does not constitute a function and cannot be used as an equality-preserving operation applied to both sides of an equation.This is why -1 = -1 does *not* imply that sqrt(-1) = sqrt(-1) ;After all, i^2 = -1 and, therefore, it is a complex square root of -1 (qualifying to be on the left side of the second equation),;At the same time as (-i)^2 = -1 and, thus, is also a complex square root of -1 (and could be on the right side of the second equation).Evidently, -i = i is false, as they are distinct complex numbers.It is worthy noting:The definition of a complex root could be restricted to make it a function.It could be that *the* complex n-th root of z is defined to be the number w with the smallest argument (the angle theta in the complex plane representation of w) such that w^n = z .This, however, is not done anywhere. Rather, the concept is defined by saying that w_i is a complex n-th root of z if w_i^n = z . And that implicates that there is more than one n-th root (for n>1). The complex root thus defined is sometimes said to be a ‘mutivalued/plurivocal function,’ in textbooks.(This is usually carried on to extend the concept of function, and does not mean that the complex root is a function proper. It is a case of ‘abuse of language.’ See https://en.wikipedia.org/wiki/Abuse_of_notation )Thus, when you write sqrt(-1) and look down at it, in the paper, no matter how suggestive it is of being ‘a number,’ it is not.It is but an abuse of language by which you are representing different numbers with one single graphical symbol. And therefore it cannot be used in equations in that same manner that you use ‘x’ , normally.While x represents, in the context of the equation, only one number; sqrt(-1) represents more than one number and therefore, sqrt(-1) = sqrt(-1) is wrong.Abuse of notation can be used effectively to produce correct results, if communications, or reasoning, take place between two parties that are well informed and conscious of the limitations of the notationally abused language. What is gained, many times, is a more compact way of expressing ideas well known to both parties.For instance, it can be economical to write and think 1/0 = infinity, or, lim (1/x) as x->0 = infinity, even though, strictly, both of these ’equation-like’ arrays of symbols have no meaning.If both parties are well trained in the definition and properties of limits, the shorthand notation can be used to perform calculations correctly, saving a great deal of paper and time needed to write formally correct statements regarding the (non-existent) limit of 1/x as x approaches 0.The danger here, is uninformed third-parties reading those and due to lack of training, drawing false conclusions.

Even understanding it all, it is not uncommon, from time to time, for one to come back to those false implications/proofs using sqrt(-1) and scratch one’s head, while going ‘damn… I know this is false, but, why, again?’ Notational intuition is a powerful force within your mathematical brain. (And it shows in the history of mathematics, in episodes where bouts of prolific development in theory come right after some better notation gets introduced.) You might wanna memorize a formalist mantra designed to safeguard you against this particular situation: ‘Sqrt(-1) is not a specific number. It cannot be hurled around inside equations just like x.’ Repeat ad nauseam until all thought is drowned. :p Put this formalist mantra in your personal mantra library along with ‘Thou shalt not divide by zero.’ and also, ‘The square root of a^2 is modulus of a, and *not* just a.’ (which helps you avoid the common and tempting mistake of assuming that a^2 = b^2 implies that a = b ) Cheers, Fabricio
Ignis 5 Nov 2023 3:25 am: X≠Y, but 0=0 so X=Y 🤪

Logical fallacies

Wed, 10 Jan 2001 12:00:00 +0000

Stephen’s Guide to the Logical Fallacies – a comprehensive list of common fallacies.