Surprised

Training a model is just rolling a ball downhill on its error. The slope says which way to go; the learning rate says how big a step to take. Too big and it flies out of the valley.

A classifier draws a line between two kinds of thing. Everything on one side is an apple, the other an orange. That line is the whole model.

A neuron adds its inputs times their weights and fires if the total clears zero. That is the atom of every neural network, and it can learn simple gates on its own.

One neuron can only draw a straight line, so it cannot solve XOR. Add a hidden layer and it can bend space until a boundary fits. Backpropagation is how it finds the bends.

Between layers each neuron passes its total through a curve that decides how much to fire. Without that bend, stacking layers just gives you one big straight line.

With no labels at all a machine can still find groups: guess some centres, colour each point by its nearest, slide the centres to the middle, and repeat until nothing moves.

The simplest classifier there is: to label a new point, just ask its closest neighbours to vote. More neighbours means a smoother, calmer boundary.

A model too simple misses the pattern; one too complex memorises the noise and fails on anything new. Good learning is the honest middle, and you can watch it happen.

Softmax turns a row of raw scores into probabilities that add to one. Temperature decides whether the top choice dominates or the whole field stays in play. Every model ends this way.

Attention lets each word look at all the others and pull in the ones that matter. It is how a model works out that "it" means the tiger, not the deer.

A language model does exactly one thing: guess the next word from the words so far. String those guesses together and it writes. This one learned from a few nursery rhymes.

Rather than re-read every word like attention, a state-space model like Mamba keeps one running summary and updates it as each word arrives. Memory decides how far back it remembers.

Stack layers and widen them and the parameter count explodes. More parameters can do more, but cost far more to train and run. This is why big models are big deals.

Words become points in space, arranged so similar meanings sit close. You can even do arithmetic with them: king minus man plus woman lands right on queen.

Every light is wired to its neighbours, so turning them all off is really a puzzle about even and odd.

Moving a tall stack is the same problem one disc smaller, done twice. That is recursion you can watch.

Fifteen tiles and one gap. Order has to be built move by move, each of which looks like a step backward.

A fully solved game. The winner is fixed in advance by the exclusive-or of the piles, not by luck.

Each guess is an experiment, and the feedback shrinks a space of over a thousand codes toward one.

AND, OR, XOR. The whole of digital logic is a handful of these little truth machines wired together.

A classic planning puzzle. Some states are traps, and the only way through is to sometimes carry things backward.

Every move must keep a rule true on both banks at once. That is constraint satisfaction, the idea behind scheduling and Sudoku.

With a 5 and a 3 litre jug you can measure a 4. The reachable amounts are exactly the multiples of the two sizes’ greatest common divisor.

You can push a box but never pull it, so one careless shove into a corner can doom the level. Real planning means thinking before you move.

A tiny Sudoku. Each row, column, and box must hold every digit once, and each filled cell rules out choices for the rest.

Cross out every multiple and the primes are simply whatever is left standing. The trick is 2000 years old.

Instant recall of the multiplication table is a muscle. Here are thirty seconds of reps for it.

A fraction is a slice of a whole, and many different fractions name the very same slice.

A straight line is a slope and a height. Threading one through a cloud of points is what regression does.

You can feel roughly how many things are there without counting them. That sense sharpens with practice.

Add, multiply, square, or Fibonacci. Spotting the rule behind a run of numbers is the heart of the thing.

Degrees measure turning. Ninety is a right angle, and every target sits at one exact angle from the corner.

The derivative is the slope of a curve at a single point. Where the slope is zero you are at a peak or a valley, which is exactly where training stops.

The area under a curve is a sum of many thin rectangles. Add more and the estimate homes in on the true value. That is integration.

A dot product measures how much two arrows point the same way: positive when aligned, zero at a right angle. It is the core of attention and of embeddings.

A 2 by 2 matrix bends the whole plane: rotate, stretch, shear, flip. Its determinant is how much area grows. Every neural layer is a matrix like this.

Gravity pulls down at the same rate whatever the sideways speed, and that is exactly what bends the path into a parabola.

An orbit is just falling forever and always missing. Too slow and you crash in; too fast and you escape.

A pendulum keeps time by its length alone. The weight of the bob never enters the period; longer just means slower.

On a slope the useful part of gravity is g times the sine of the angle. Steeper means a bigger push down the ramp.

When two equal balls hit, they swap the speed along the line between them. Energy sloshes around and barely leaks away.

What tips a lever is not weight but weight times distance from the pivot. That product is torque.

A spring pulls back in proportion to how far you stretch it. Let go and it swings at a rhythm set by its stiffness and the mass, never by how far you pulled.

In an elastic collision total momentum is exactly preserved. A heavy cart barely flinches when a light one hits it; two equal carts swap speeds.

Holding a growing sequence in your head is working memory, stretched one extra step every round.

Where was that pair? Memory for places improves surprisingly fast with a little play.

From a flash of light to your click is about a fifth of a second. Here you can measure your own.

Reading is so automatic that naming the ink of a colour word fights the word itself. That clash is the Stroop effect.

Every whole number is a sum of powers of two, which is all a computer ever counts in. Eight switches reach 255.