Moontower #173

Let’s start with a math puzzle from Martin Gardner’s Entertaining Mathematical Puzzles:


A silver prospector was unable to pay his March rent in advance. He owned a bar of pure silver, 31 inches long, so he made the following arrangement with his landlady. He would cut the bar into smaller pieces. On the first day of March, he would give the lady an inch of the bar, and on each succeeding day he would add another inch to her amount of silver. She would keep this silver as security. At the end of the month, when the prospector expected to be able to pay his rent in full, she would return the pieces to him.

March has 31 days, so one way to cut the bar would be to cut it into 31 sections, each an inch long. But since it required considerable labor to cut the bar, the prospector wished to carry out his agreement with the fewest possible number of pieces. For example, he might give the lady an inch on the first day, another inch on the second day, then on the third day he could take back the two pieces and give her a solid 3- inch section.

Assuming that portions of the bar are traded back and forth in this fashion, see if you can determine the smallest number of pieces into which the prospector needs to cut his silver bar.

Don’t read further unless you want the solution.

This is the solution:

The prospector can keep his agreement by cutting his 31-inch silver bar into as few as five sections with lengths of 1, 2, 4, 8, and 16 inches.

You’ll note that you can sum to any length up to 31 with that set of bars.

Disclosure: I read this problem aloud to my kids on Wednesday night at bedtime and none of us got the answer.

The key to this problem is it can be expressed cleanly using binary or a base-2 number system instead of using the base-10 system we are used to.

Earlier that afternoon I taught Zak (9) how to count in binary. I made these charts to summarize how I explained it to him.

It’s useful to remember that when you hear “tens place” what is really meant is the [symbol occupying that space] x 10¹

  • In base-10: there’s a one’s place, tens place, hundreds place…
  • In binary: there’s a one’s place, twos place, fours place, eights place…

Once Zak understood this he could represent our family members’ ages in binary. I had him do the same thing using base-8 by following the same pattern.

  • In base-8: there’s a one’s place, eights place, sixty-fours place, five hundred and twelve place…

So representing my age of 44 in base-8 = 54 because there are 5 eights + 4 ones.

Computers do calculations in binary because electrical signals on a chip are either on or off. So representing numbers can be thought of like a morse code of “on”/”off” at light speed or something. Feel free to correct me, that’s my armchair impression.

Now a brutal coincidence.

Despite teaching binary that afternoon, I did not recognize it was the key to answering the riddle! Zak was flabbergasted by the coincidence that the solution was in “what’s that word you keep saying…binary?” after he just learned it. Baader-Meinhoff on steroids. (It was really cute that he kept shaking his head about the coincidence that the thing he just learned showed up in a random problem I chose that night. The coincidence hit him just as hard, maybe even harder, than it did me.)

The silver bars cut into 5 lengths as a binary solution:

This is all about to get better (or worse depending on what temperature you run at on the techno-optimism thermometer).

Yesterday, I started tinkering with OpenAI’s recently released ChatGPT. I found this concise description of ChatGPT:

A large language model trained by OpenAI is a machine learning model that has been trained on a large dataset of natural language text, typically using a deep learning approach. This allows the model to learn the patterns and structures of human language and to generate text that is similar in style and content to human-generated text.

You can sign-up to try it for yourself here.

This is a Twitter thread of prompts I’ve given it and its responses. Everything from a Kelly Betting problem to imagining if DMX wrote the hook for No Scrubs.

I also gave the bot a condensed version of the riddle.

Prompt: If I have a 31 inch silver bar but I need to make the minimum number of cuts so I can create a new bar any length up to 31 inches, how many cuts do I need?

It responded:

To create a bar of any length up to 31 inches, you would need to make five cuts to your silver bar. This is because the minimum number of cuts required to create a bar of any length up to a given length is equal to the log base 2 of that length, rounded up to the nearest whole number. In this case, the log base 2 of 31 is 4.8, which rounds up to 5. Therefore, you would need to make five cuts to your silver bar to create a new bar of any length up to 31 inches.

“log base 2”?! It knew it was a problem for binary.

Lacking the vision to peer around corners, my small mind can only see the 6 inches in front of my face. And I can see turning to ChatGPT over Google for many types of searches. But with the explosion of “generative AI” in the past year (GPT engines, Dall-E, Stable Diffusion or Midjourney for images), it’s clear the times they are a-changin’

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Money Angle

Author and veteran trader Brent Donnelly’s recent post, #37, in his highly educational, free Substack series 50 Trades In 50 Weeks dissects common traits of successful traders:

Week 37: Common Traits of Top Traders (10 min read)

It’s a terrific read for what it validates, but also for what is surprising (“You can succeed in trading with any level of risk appetite.”) It’s also a great read to see how he constructs a survey to challenge his beliefs.

I want to zoom in on one of the key 5 key traits he identifies: “calibrated confidence”.

When we hosted the StockSlam sessions in October we gave a couple of homeworks before the in-person events. One of the homeworks involved this module:

You will be asked to make 90% confidence intervals on some facts.

90% confidence interval can be understood with an example:

I have not been outside for a few hours, and without looking up the answer, “I’m 90% confident that the temperature outside my window is between 45°F and 60°F, or (45,60).”  If I wanted to be more (like 99%) confident, I would widen the interval, and conversely a tighter interval would coincide with less confidence.

Example questions included:

  • Without looking anything up, what is your 90% confidence interval on the number of bones in the human body?
  • What is your 90% confidence interval on the length (# of letters) of the longest word in English?

Note what you are doing. You are making markets where you think fair value is 90% to be inside your bid/ask spread.

This is highly relevant to trading and handicapping.

If you get 10 out 10 markets “right” then you are more conservative than we asked you to be. In other words, you were too wide. In markets, this means you will never trade. You are bidding $50 for a stock trading $60. You will get no market share.

If you get say 5 out of 10 markets right, then you were too confident. In markets, everyone will trade with you and you will be sadder for it.

Trading is only partially about knowing “fair value”. It’s in your meta-knowledge about fair value that the magic happens. You are always dealing in uncertainty…your feel for the degree of uncertainty is how you recognize opportunity or defend yourself. This is not just market-maker talk. It’s the essence of what Buffet understood when he said “margin of safety”.

Google “Paul Slovic’s horse bettor study” and you will find several posts such as:

How Can Confidence Kill Investment Returns? (4 min read)

They all talk about a 1973 study where experienced horse handicappers are given a few pieces of data of their own choosing. Armed with their preferred data, they are able to make good bets, but critically, they are well-calibrated about their accuracy. Their confidence and accuracy were in agreement.

However, as the bettors are given increasing amounts of data their accuracy falls, but their confidence shoots up. No bueno.

Presumably, they were less experienced in weighing the additional data which turns out to be noise to their handicapping process, but the presence of more info gave them the illusion that they would be better. A disastrous combo.

Improving Our Handicapping Skills

It sounds discouraging to realize that decision-making is not just an exercise in accurate prediction but also judging how wide our error bars should be. But there’s plenty of good news. This is not a skill anyone is born with. It is learned.

[This is why prop trading firms who recruit elite students from top schools screen for teachability which I suspect is randomly allocated across skill. There are elite students or athletes who can remain humble learners despite the advantages of their talents and there are insufferably overconfident talents who have yet to get hit by the Mack truck of peak competition. This latter group is dangerous to themselves and if given enough rope, to their backers too.]

3 suggestions to get better:

  1. Self-diagnose. You can take a fun calibration test to establish some context:

    This thread includes my results as well as many others including several side threads of interpretation and discussion.

  2. Read Superforecasting: The Art and Science of PredictionThis book is a great primer on what it takes to predict but most importantly frames prediction as a skill we can improve at. I took extensive notes which weave my thoughts into some of the key points.
  3. Bet on stuffHow long will it take to finish your school or work assignment?

    What percent of random sample of people from your contact list knows what date the winter solstice falls on?

    (My wife and I bet on what facts are “common knowledge” when one of us gets defensive to the other’s attack “how did you not know that?” We’ll poll some people with a text message and bet on the consensus. These “studies” are highly correlated with one of us having “one drink too many”.)

    Calibration is a nice benefit of a betting culture (1 min read).

Last Call

I saw this tweet by one of my favorite accounts:

In light of the extreme timeline we seem to be living in these days, I’ll share a spot-on observation from Rohit Krishnan’s most excellent post How VCs Can Avoid Being Tricked (emphasis mine):

Don’t fall in love [with founders] more than necessary:

Try to internalise the following: “human ability is normally distributed but the outcomes are power law distributed”. What this means is that just because someone builds a company that produces extraordinary outcomes, 10000x the average, doesn’t mean that they were 10000x as capable. Achievements are created from multiplicative outcomes of many different variables. So if you’re investing in a “10x founder” it doesn’t mean that they themselves are 10x the capability of everyone else, but what it means is that their advantage, combined with everyone else’s advantage, can get you to a 10000x outcome. Which means the adulation we pour on top of some folks creates its own gravitational field, and makes others susceptible to falling in love. The most difficult task is to not let someone else control your decision making for you, which is what you give up. If your job is to get seduced by the right narrative by the right-seeming person, guess what you’ll get seduced by anyone who can tell a compelling narrative.

I think the advice, written in an investment context, can be generalized to apply to Visa’s concern.

From My Actual Life

I hate shopping. I don’t pay attention to Black Friday or Cyber Monday. Until this year. I just went bonkers and spent all of Monday incinerating cash while feeling exactly what the gods of retail want me to feel like — “I’m getting a bargain!”

This might help your gifting:

How To Become A Truly Excellent Gift Giver (9 min read)
by Eliza Brooke

Anyway, I’m in charge of buying all the kids in our lives gifts and no parent wants clutter, so books it is.

crowdsourced books for kids 9-12 and compiled the results in a table which I will add to over time.

Book Ideas For Kids

I’ll add a special mention for Rookie Bookie which I read with Zak this year. It’s written by a Wall Street quant with a sports analytics love. This observation from the book is a nice demonstration of how the book is fun for the adult reading with the kid:

Announcers say a team has won 3 out of the last 4 because it sounds more impressive than saying they won 3 out of the last 5. Even though that’s actually what happened.

Another famous Wall Street quant and author did a better book review than I could:

Last thing…Moontower reader Murph got inspired by my incessant advocacy of Notion combined with to quickly turn your into a website. Here’s what he built:

Never been easier to scratch your generative itches folks…

Stay groovy!

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