Naismith's Rule provides a quick estimate for hiking time in the mountains, intended for "easy expeditions." It suggests:

+1 hour for every 2,000 ft (600m) of ascent

For example, suppose your hike covers 6 miles with 2,000 ft of ascent. In that case, Naismith's Rule suggests allowing 3 hours total—2 hours for distance and 1 hour for elevation.

(Here's a metric sketch of Naismith's Rule)

William Naismith was a Scottish Mountaineer born in 1856. He spent years trekking in the Highlands and was, to be fair, probably made of nails. In the Scottish Mountaineering Club Journal in 1893, he wrote:

"...a simple formula, that may be found useful in estimating what time men in fair condition should allow for easy expeditions, namely, an hour for every three miles on the map, with an additional hour for every 2,000 feet of ascent."

The Scottish Mountaineering Club Journal, Vol. II, September 1892, No.3, p.136 (pdf)

A figure I learned that's stuck with me is that a brisk walk is around 4 mph. But that's actually pretty fast and wouldn't apply on a mountain trail and certainly not hiking up a mountain. So I've run into the challenge myself when planning a trip in the mountains of how to guess how long a hike will likely take.

From a recent hike—where I made sure to run through the full kitchen table survival skills and 10 essentials before leaving—limited daylight was a safety consideration. So knowing how long the hike might take was really important. Incidentally, there's a rule of thumb, literally, for estimating remaining daylight.

Lots of people have put Naismith's Rule to the test, including me, and though it's a slightly optimistic metric, it's a starting point. On my recent hike, we were slightly over, but not by much.

To adjust for your circumstances, add time for:

Tricky or technical terrain and trail condition

Not being "in fair [physical] condition" in the view of a rugged mountaineer from Scotland in the 1890s

Photography, bird watching, animal spotting

The size of your group—I have heard the rule of thumb to add 15 mins for each person in your group

Presumably, if the hike was really long, you'd slow down

So, there are many reasons why Naismith's Rule may be on the light side. But it's a decent place to start from.

Killian Jornet Laughs in the Face of Naismith's Rule

Someone for whom Naismith's Rule definitely doesn't apply is the incredible Killian Jornet, whom I watched in the beautiful documentary Déjame Vivir on Netflix. Footage of him running along knife-edge ridges at the top of the Alps is breathtaking. In 2024, he hiked all 82 peaks above 4,000m in the Alps in 19 days without using motorized vehicles between them 🤯.

I was told about the term lock-miles, a kind of Naismith's Rule but for canal journeys rather than hikes. To estimate the journey time, you'd first calculate the lock-miles with the idea that it takes as long to travel through a lock as it does to travel a mile.

A journey of 10 miles with 5 locks would be 15 lock-miles. For experienced boaters, a typical speed might be 3 lock-miles an hour, so the journey of 15 lock-miles would take 5 hours.

Let me know if you give Naismith's Rule a try!

Prefer metric? So do I. Here's a metric Naismith's rule (with the approximate rounding)

Kaffikok — the time you can travel before needing another cup of coffee

I made some prints and products with and without the text in case, like me, you like a peaceful mountain ridge without any rules

Naismith's Rule for mountain hiking time

Nearly every iceberg you see in a picture or diagram is probably floating the wrong way. This was what I learned (from Megan Thompson-Munson) after sketching Biz Stone's brilliant saying about the myth of overnight success.

With some approximation, the density of ice is around 900 kg/m3, and seawater is around 1,000 kg/m3. Therefore, the fraction of an iceberg that's submerged is around ~900/1000 or 0.9. So, about 90% of an iceberg is below the surface and 10% above, which is partly why they can be so dangerous. While most iceberg pictures get this part more or less correct, most of these icebergs will be floating vertically. In reality, a tall, thin iceberg will likely topple, so most icebergs end up floating on their side, not their tips, even though we rarely draw them this way.

I remember learning about a fascinating experiment with children of different ages estimating which glass holds more water: a tall, slender one filled high or a wider glass filled to a lower level. Younger children almost always chose the glass with the higher water level as the most water, even when it was significantly less than the shorter and wider glass. I wonder if it's part of why we draw icebergs vertically, at least when we're using them as a metaphor. It's easier to grasp quantities vertically, and we commonly underestimate volume spread over a wider area.

We may still want to draw our icebergs tall and deep to make our point, but now, at least, we can do so with the knowledge that they're not like the real ones.

If you want to see it yourself, Joshua Tauberer made a brilliant draw-an-iceberg-and-see-how-it-will-float game. I recommend you give it a go so it sinks in forever (sorry).

Know your poles: penguins or polar bears, frozen ice or land

Iceberg orientation

Parallax is the change in the apparent position of objects from different viewpoints. For example, from one viewpoint, a nearby house may appear in front of a hill, and from another, it may be in front of a lake. Objects at different distances appear to move by different amounts. As you move between viewpoints, say when driving along a road, nearby objects move past you quickly, while distant objects appear to move by slowly. This difference in motion from near to far objects helps us determine how far objects are away.

I remember learning parallax from video games, like playing Sonic the Hedgehog on the old Sega Mega Drive. As the character moves in the foreground, elements at different distances in the background move progressively slower, creating a sense of depth. Saturation and contrast change at with distance also, known as atmospheric perspective.

I also enjoy observing parallax while looking out of a train window; nearby hedges and trees race by, buildings in the middle distance move more slowly, and distant hills or mountains barely seem to change. It's fascinating how our brains interpret this relative movement to judge distances.

One of the more intriguing applications of parallax is stellar parallax in astronomy. The basic geometry of parallax allows us to measure the distance of (relatively) nearby stars by observing relative shifts against the background of distant stars. Here's how it works: we observe the stars from different points in our orbit around the Sun. As a result, the stars appear to move relative to each other. By measuring this apparent shift, we can calculate the distance of the stars.

Parallax

A recap of some of the fundamental parts of a circle, including some that are easy to forget:

Diameter — a line crossing the circle through its centre. Also the longest straight line you can draw in a circle.

Radius — a line from the centre to any point on the edge. Also, half the diameter.

Sector — the classic Trivial Pursuit cheese, pizza or cake slice formed by two radii.

Area — equal to pi x the radius squared.

Circumference — equal to pi x the diameter.

Chord — a straight line touching any two points of a circle. The longest chord is the diameter.

Segment — the areas on either side of a chord, major and minor.

Tangent — a straight line just touching the edge of the circle without crossing it. It forms a right angle to the radius on the edge. Just as if you were able to lean a ladder against one.

More maths and science refreshers: States of matter, BODMAS (or PEMDAS), the 9 times table on your fingers, binary, incidence and reflection

Parts of a circle

There's an old story about the statistician who drowned after seeing that the average depth was 3ft. Averages, or in this case the mean, necessarily hide some data, but very often they also hide what's really going on.

Let's say you run a delivery service and have an average delivery time of one day, it could be that most deliveries are actually a few hours while a few particular ones have people waiting for a week. Or a figure of average incomes might hide that most people have low salaries and a few are millionaires. But just looking at the average you can't tell. Sometimes a different measure of central tendency, like the median, can provide a clearer picture.

Sheldon Zedeck, a UC Berkeley psychology professor who taught me about the design of experiments, gave wise advice: spend time with your data. Sometimes it's the only way to know what's happening for sure.

Sneaky averages

Möbius strip: the mindbending shape with only one side.

It's very simple to create: take a long piece of paper, give one end a 180-degree twist and then stick the ends together. The simple shape just created has only one side. What??

You can test this by running your finger over the surface and you'll cover the entire shape and end up back where you started.

Try to follow the red ball in the animation as it follows the surface over the entire loop.

Möbius strips have a couple of other interesting properties including how cutting them in half down the middle simply makes a longer loop, and cutting a third of the way in over the full length will reveal two distinct yet interconnected loops. Fun to try!

I once spent a memorable evening with a friend trying to feed a Möbius strip through a regular printer to see if we could print on both sides in one go.

SP 759 - Möbius strip - single.png

In case it's handy, here's a static Möbius strip sketch

Möbius strip

Brilliance bias is the tendency for people to think of 'brilliance' as generally a male trait. With fewer women generally recorded in the history books, the brilliant people that come to mind from a young age tend to be men.

Underrepresentation has consequences. Careers that are generally understood to require 'brilliance' to succeed, such as maths, physics, or philosophy, are likely to have fewer women working in them. Even at school age, attitudes start to shift in both girls and boys. For example, having an interest in games that require you to be 'really smart' or whether, when asked to draw a scientist, we draw a woman or a man. We need to stop teaching brilliance bias, and we need to celebrate the many female role models we have.

My sister shared with me a page on drawing famous faces that showed 49 men to draw, and an example of the board game Guess Who having as many women to guess as it has men with facial hair as an attribute.

Caroline Criado Perez's eye-opening book Invisible Women taught me about brilliance bias, including the examples here.

Brilliance bias

School mathematics refresher: when faced with a calculation that looks like it could be tackled multiple ways, use BODMAS to help clear it up (See PEMDAS at the end).

BODMAS is an acronym (as opposed to an initialism) that tells you the order of operations to tackle in a calculation.

BODMAS stands for Brackets, Order/Indices, Division, Multiplication, Addition, and Subtraction. So in 2 + 4 x 2 you don't end up with 6 x 2 = 12 but instead, 2 + 8 = 10

BODMAS is generally built into calculators (you can search 2 + 4 x 2 in Google to see). If you need additions or subtractions to be carried out first you can surround them in brackets eg (2 + 4) x 2

I always learned it as BODMAS, but many people told me they learned this as PEMDAS. The idea is the same just the words are different, and the ordering of Division and Multiplication, which are interchangeable:

PEMDAS: Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction

BODMAS

If you wanted to know who would win an election, then the only reliable (most of the time) way to do that is to ask everyone to vote for real and count them all. Short of doing that, if you want to get an idea who would win you can pick a group that you hope is representative of the whole of the voting population, ask them what they plan to vote instead, and then multiply up to the full group. But it turns out, whenever you ask people things, it's jolly hard to really have confidence in what you're extrapolating up to as all sorts of bias can creep into your sample.

If you ask people what they plan to vote they may lie, they may tell you what they plan to do then change their mind, they may tell you what they think you want to hear depending on who's asking, they may say the first option because it's easiest, and you may ask in the places where it's easy or cheap to do so, or hear from only the people who are happy and not too busy to talk with you.

In the example in the sketch, if you hope to receive responses to your questions by surveys then you get responses from people who happen to not mind doing surveys — from anyone who throws them in the bin you won't receive a single data point.

All of these, and others, can lead to a bias in your sample and, at times, to drawing wholly inaccurate conclusions.

Extreme example from How to Lie with Statistics by Darrell Huff.

Sampling bias

We find it handy to count in the decimal system using 10 numbers from 0,9, known as base 10, before we have to put two together to make 10 to keep counting further — 10 fingers and toes and all. But it turns out you can represent all numbers equally using just two digits, known as base 2, a 0 and a 1. This is called the binary system and is credited to Gottfried Wilhelm Liebniz in the 1600s.

The binary system is handy because 1s and 0s can be represented by simply by on/off and reproduced in as simple means as pebbles in trays , the sign of a magnetic field eg positive/negative, or a gate as open or closed. This made them the choice for designing computers and is how digital information is stored and transmitted today.

When we write a decimal number we use positional notation with each successive position representing 10 to the next power. So 739 is understood as (7 x 10^2) + (3 x 10^1) + (9 x 10^0). 10 to the power 2 is 10 x 10, so is the number of 100s in the number. Anything to the power 0 is 1.

The same is true in binary, so 1101 is understood in decimal as:

1101 = (1 x 2^3) + (1 x 2^2) + (0 x 2^1) + (1 x 2^0)

1101 = (1 x 8) + (1 x 4) + (0 x 2) + (1 x 1)
     = 8 + 4 + 0 + 1
     = 13

Each 0 or 1 in a binary number is known as a bit — named by Claude Shannon as short for binary digit — and 8 bits is known as a byte.

Translating to decimal looks fiddly, but computers don't have to do that, they can just add, subtract or multiply the binary numbers directly. Amazing to think that the device you may be reading this on now is just incredibly efficient at manipulating 1s and 0s.

For a readable and visual introduction to the history and operation of computers — from binary, logic gates, transistors, circuits, and Moore's law through to software and AI — you could do a lot worse than my Dad's book The Computing Universe 😀

Binary

It might seem straightforward to pull out a map and measure the distance between two distant points, however, the larger the distance the bigger the distortion caused by traveling on the curved surface of the Earth as opposed to flat 2D space. So while the distance you measure to your neighbouring town won't be too bad, if you're measuring between London and Rio the curvature of the Earth will make a big difference to the distance that you'll travel. To help figure out the correct distance there's the haversine formula.

The haversine formula allows you to calculate the shortest distance between two points on a sphere using their latitudes and longitudes — this will be the arc between them on the great circle that includes both points. A great circle is a circle on a sphere with the same centre as the sphere, like the Equator. The haversine formula isn't perfect in practice, as Earth isn't a perfect sphere — see the 3 tallest mountains.

It was invented around two hundred years ago, together with tables to speed any calculations, to help sailors navigate.

The haversine formula

You may remember this little trick from school. If you’ve not seen it before, give it a quick try 🙂

The 9 times table on your fingers

The coastline paradox is the fascinating observation that it’s not straightforward to say how long a coastline is. If you were to measure the coastline of a country by using a ruler on a globe, you would come out with a vastly different number than if you were to pace around the edge. The closer you look, the more wiggles and squiggliness you come across, and instead of converging to a more accurate length, the coastline just keeps getting longer. The smaller your ruler, the longer it gets.

This was originally spotted, incredibly, in the 1950s by an Englishman, Lewis Richardson, when trying to check a theory he had that the likelihood of war between countries depended on the length of their shared borders. Remarkably, he found that the quoted lengths of borders varied significantly. While measuring on maps at different scales, he saw that the smaller scale map he used, or the smaller the width of his callipers he measured with, the length systematically increased. When looking at coastlines, instead of borders, some countries had wigglier coasts, and so the length increased at a faster rate with the scale — for instance, Norway’s coastline, with it’s crinkly fjords, increases faster than Britain’s, which in turn increases faster than South Africa’s, as he zoomed in. The rate of this increase later became known as its fractal dimension.

Long after Richardson’s research, Benoit Mandelbrot published a paper How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension that discussed how the wiggliness of something like a coastline at one scale can be repeated at smaller and smaller scales. The work led to the later term fractals. Many other things exhibit fractal-like behaviour, such as river networks, borders, brains, frequencies, lightning, and even the stock market.

There’s a super section on this in Scale, by Geoffrey West.

The difference between Great Britain, the UK and the British Isles

The coastline paradox

In my everyday life I’d have been hard pushed to put my finger on the difference between accuracy and precision but, in engineering at least, you learn that there is a clear difference between the two.

Precision describes how variable an outcome is, or how close repeated measurements or actions are to each other.

In contrast, something may be very precise but not be accurate to your intent or to the true value of a thing. That’s accuracy — how close you are to the true or intended value. If you’re hitting near your intended target then you’re accurate.

Hmm…one of those things where a sketchplanation works a whole lot better than words.

For the more spiritually inclined: Everything is aiming

Or for the more statistically inclined: Measures of central tendency (mean, median, mode), and Common distributions (normal, skewed, pareto)

That accuracy is true to intention and precision true to itself, I learned from Simon Winchester in Exactly: How precision engineers created the Modern World.

Accuracy and precision

Since I learned this remarkable trick I actually find I quite like getting a wobbly table for the chance to retest this simple, and still kind of remarkable, fix. A number of smart people have spent probably more time than justified demonstrating that, for a four-legged table with even length legs on uneven ground, as long as the ground is not excessively bumpy you’ll be able to fix your wobble by rotating the table up to ¼ turn.

Now, it’s definitely true, that if you have a square table, not a round one, or, say, two square tables next to each other, it’s not always possible to start rearranging the café, but…when you can it’s really neat to see it work. Give it a try.

Fix wobbly tables by rotating

Idempotence is essentially an operation that, no matter how many times you do it, you’ll still get the same result, at least without doing other operations in between. A classic example would be view_your_bank_balance being idempotent, and withdraw_1000 not being idempotent. It’s a property that’s often handy as you can retry the operation without worrying about unintended effects.

Idempotence is a concept from mathematics and computer science but can be applied more generally. For example, simple on and off buttons are idempotent, a property that can be very useful in emergencies.

Idempotence

The proportions of the golden ratio pop up in all sorts of places. They are a handy shortcut to make something we seem to innately find pleasingly proportioned.

In nature, the golden spiral abounds in the shell of the nautilus, the spiral of strawberry seeds, the spikes of cacti, and hurricanes.

It’s found in the pyramids, the Parthenon, the Mona Lisa, violins, photography and, very deliberately, in corporate logos. No wonder it’s sometimes called the golden ratio of beauty.

There’s a pleasing simplicity to the equation too, where the ratio of the long over the short dimension is the same as both added together over the long dimension.

Check out some golden ratio examples. Even Trump.

This sketch features in my book Big Ideas Little Pictures along with 136 other fun and fascinating concepts.

The golden ratio

Loosely based on the game show Let’s Make a Deal, hosted by Monty Hall, what’s known as the Monty Hall problem is a simply constructed puzzle. Imagine in a game show contestants saw 3 doors, and behind one of which is a prize, and behind the other two doors, goats. The contestant chooses a door, but instead of opening it, Monty opens one of the remaining doors that reveals a goat. He gives you a choice: Do you want to change your door or stick with the one you picked? The problem is: which is the better choice?

If you’re interested it’s pretty easy to find out the real answer. Confirming to yourself why is another thing.

The Monty Hall problem

It’s a funny thing, when you think about it, that if you owned a stock in the market and the value went up 50%, and then went down 50%, that it wouldn’t end up at what it started.

Another way to look at it, is if you owned something worth 100 and it lost 50%, it’d be worth 50. To get back to being worth 100 you’d then need a 100% gain. 

Asymmetry of returns

Tactics drawn from How to Use Math to Dominate at Monopoly.

Win at Monopoly

OK, there are probably many candidates, but it is remarkable to me how such esoteric and difficult to define numbers - derived independently - have such a simple elegant relationship.

Math sketches