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

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

You have to drop off 2 things after school, get to the shops and back home — which route should you take? You drive an Amazon delivery van and you have 145 parcels to deliver across London — what’s the most efficient route?

It’s pretty clear that for situations like the first one, with fewer stops, assuming you know travel distances or travel times, you could figure out the best route by trying out the different combinations until you hit on the shortest. The trouble is as you keep adding extra stops it doesn’t get just a little bit harder — instead the difficulty continues to increase along with the time it will take to find an answer.

Figuring out the shortest route to visit all the stops and return back home is known as the travelling salesman problem. It’s a problem formulated over 150 years ago that still has relevance and interest today whether it’s for delivering parcels, stocking shelves or soldering transistors.

The travelling salesman problem

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

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 golden ratio is a mathematical proportion in nature, art, architecture, and design. It's a shortcut to proportions we seem to find innately pleasing.

The golden ratio has an elegant mathematical definition: the ratio of the long over the short dimension is the same as both added together over the long dimension. It works out at around 1.618. This relationship creates a golden rectangle.

If you remove a square from a golden rectangle, the remaining space is another golden rectangle. This process can continue indefinitely, forming the golden spiral.

In nature, the golden spiral abounds. Find it in:

The shape of some seashells and the growth patterns of cacti

Designers, artists and architects have used the golden ratio intuitively and intentionally for 1000s of years. So you can find the golden ratio in:

No wonder it's sometimes called the golden ratio of beauty.

The golden ratio is a tool for balance and harmony. Using it in your craft—whether creating graphics, taking photos, or designing a website or building—can bring balance and harmony to your work.

You can use the golden ratio to guide the layout of elements.

Use the golden rectangle: Divide your design area into a golden rectangle. This gives you a framework for positioning your text, images, and other elements in a visually pleasing way.

Spiral composition: Divide your golden rectangle into a golden spiral. Position key elements of your composition where the spiral leads for a flowing, natural composition. The spiral works well for web design, posters, or advertising, leading the viewer's eye through your content.

Proportions and spacing: When designing logos, posters, or even UI layouts, using the golden ratio for the size relationships between text, images, and borders can create a sense of proportionality that feels balanced.

You can also make use of the golden ratio in photography. Just as with the rule of thirds, you can use the ratio to frame your subjects, with focal points falling along key intersections. This can create harmony and depth in the shot.

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.

One-Point Perspective, Two-Point Perspective, Three-Point Perspective

The Golden Ratio: Nature's Perfect Proportions

Asteroids are often considered irregular, lumpy, potato-shaped rocks, while planets seem smooth and spherical. The “potato radius” is the fascinating threshold where this transformation begins—a concept that illustrates how gravity shapes celestial bodies.

When a space rock’s radius (the distance from its centre to its edge) grows to around 200–300 kilometres, its own gravity becomes powerful enough to reshape it. At this size, gravity pulls the rock’s mass inward toward its centre, overcoming its irregular structure and creating a more spherical shape. Think of it like compressing a crumpled ball of paper into something rounder and smoother.

Rocks smaller than this threshold retain their irregular, asteroid-like shapes, while larger bodies transition into planet-like spheres due to self-gravity. This natural process highlights the profound role gravity plays in shaping the universe.

Potatoes as Asteroids in The Empire Strikes Back

Our image of asteroids is so closely linked to potatoes that they have even served as asteroid doubles in movies. In a brilliant example of creative special effects, The Empire Strikes Back used potatoes as some of the asteroids in the famous scene where the Millennium Falcon flies through an asteroid field.

The potato radius and other remarkable natural phenomena feature in Professor Brian Cox in the BBC series Forces of Nature.

Not sure about Asteroids, meteors and meteorites?

The potato radius features, along with many other fun, spacey-sketches in my book Big Ideas Little Pictures

The Potato Radius: When Asteroids Turn Spherical

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

The time to a target depends on the ratio of distance to the target over it’s size. So clicking on a small target close by might take a similar time to clicking on a large target further away. It’s a simple model and there are a few little games out there to bring it home in practice.

I’m reminded however, of an observation from a mentor of mine, Darrell Mann, that just because people call it a law doesn’t mean you have to treat it like one. The example of Apple’s standard program icons at the base of the screen that expand when you mouse near them, are a simple technique that help to overcome the limitations of Fitt’s law. There are plenty more. Don’t be sucked in to laziness by a name.

Fitts’ Law

Very easy to forget the difference between the mean, the median and the mode.

Median: the middle value of a list in ranked order.

Measures of central tendency: mean, median, mode

Normal (the weight of apples, measurement errors, IQ…), skewed (Number of children in a household, age at retirement…) and Pareto (80/20 rules, long-tails, size of meteorites).

Common distributions: normal, skewed, pareto

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.

Mathematics sketches