DRY — Don't Repeat Yourself: Visual walkthrough

The parent note DRY — Don't Repeat Yourself made a claim and then dropped two formulas on you:

$C_{edit} = n \cdot c P_{consistent} = (1 - p)^{n}$

This page builds both from absolute zero — no symbol appears before you can see it. By the end you will feel, in pictures, exactly why one copy is safe and five copies are a time bomb.

Let us first agree on what every letter is going to mean, one at a time, as pictures.

Step 1 — What is "a copy"? (setting the scene)

WHY start here? Every formula below counts copies. If we do not nail down what we are counting, the maths is meaningless. So the very first symbol we earn is:

PICTURE. Below, the same fact "TAX = 18%" is drawn as identical sticky notes. Count the notes — that count is $n$ .

Look at the yellow notes: there are five of them, so here $n = 5$ . Nothing more complicated than counting is happening yet.

Step 2 — What does it cost to fix ONE copy?

WHY invent this number? Because we want to compare "editing one place" with "editing many places". To compare, we need a unit of effort for one place. That unit is:

PICTURE. One sticky note, one hand editing it, one little cost-bar of height $c$ .

The single blue bar is $c$ : the price of touching one home. Remember its height — in the next step we start stacking these bars.

Step 3 — The total edit cost: stacking the bars

WHY multiplication? "Add $c$ to itself $n$ times" is exactly what multiplication means. So the total is $n$ lots of $c$ :

Each symbol's job:

$n$ — the count from Step 1 (how many bars we stack).
$c$ — the unit from Step 2 (how tall each bar is).
$C_{edit}$ — the total height of the stacked bars: the whole bill.

PICTURE. Watch the bars stack as $n$ grows: $1 c, 2 c, 3 c, \dots$ — a straight ramp.

The dots march up a straight line. This is what "grows linearly" means: double the copies → double the bill. No surprises, just a steady, honest tax on every extra copy. (The dangerous part is coming in Step 5.)

Step 4 — What is the chance you MISS a copy?

WHY a probability? Because "forgetting" is not certain, it is a risk. Risks are measured as probabilities (0 = never, 1 = always). So we earn:

PICTURE. A single coin split into two slices: a $p$ slice (forget) and a $1 - p$ slice (remember).

The blue slice $1 - p$ is "I handled this copy correctly". That single slice is the seed of the whole exponential decay — because now we need every copy to land on blue.

Step 5 — All copies correct: why the chance collapses

WHY multiply the $(1 - p)$ 's? When independent events must all happen, their probabilities multiply. (Chance of two coins both landing heads $= \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4}$ .) Getting each of the $n$ copies right is $1 - p$ , and they must all succeed, so:

PICTURE. A curve of $(1 - p)^{n}$ plotted against $n$ . It starts high and dives toward the floor.

Trace the pink curve: at $n = 1$ you sit at $1 - p$ (as high as it ever gets). Each extra copy multiplies by another $1 - p < 1$ , so the height keeps shrinking — this is exponential decay. Compare it to the straight ramp of Step 3: cost grows steadily, but safety collapses fast.

Step 6 — The degenerate cases (do the formulas behave?)

A formula you trust must survive its extreme inputs. Let us test every corner.

PICTURE. The two curves side by side, with the four corner cases marked as chalk dots.

Notice the vertical dashed line at $n = 1$ : it lands cost at its minimum and safety at its maximum. There is no corner of the input space where "more copies" wins — that is the derivation's verdict.

Step 7 — A concrete number to make it real

Let $c = 5$ minutes, $p = 0.1$ (you miss 1 in 10). Compare $n = 1$ (DRY) against $n = 5$ (WET):

That single comparison is the entire argument for Single Source of Truth in two numbers.

The one-picture summary

Left panel: cost climbs a straight line ( $n \cdot c$ ). Right panel: safety plunges as a curve ( $(1 - p)^{n}$ ). The green $n = 1$ marker sits at the sweet spot of both — cheapest and safest. Every step right of it pays twice: more work and more risk. That crossroads is DRY.

Recall Feynman retelling — the whole walkthrough in plain words

Picture a fact written on sticky notes. Count the notes — that count is $n$ . Fixing one note takes some effort; call it $c$ . So fixing all of them is just $c$ stacked $n$ times, $n \cdot c$ — a steady, straight climb: twice the notes, twice the work. But you're human, so each note has a small chance $p$ of being forgotten, meaning a chance $1 - p$ of being done right. To be safe, every note must be right at once, and "everything at once" means multiplying — $1 - p$ times itself $n$ times, $(1 - p)^{n}$ . Multiplying a number smaller than one, over and over, races toward zero. So as notes pile up, your effort rises gently but your safety crashes. The only spot that's both cheap and safe is one note — one home for the fact. That's DRY: one fact, one home.

Connections

DRY — Don't Repeat Yourself — the parent; this page derives its two formulas from zero.
Single Source of Truth — Step 7 is its numeric justification.
WET — Write Everything Twice — the $n > 1$ world we just measured.
Rule of Three — remember: $p = 0$ shows duplication isn't always deadly, so wait before abstracting.
Refactoring — Extract Method — how you actually drive $n$ back down to 1.