Week 28 · Human Interaction Psychology

The Doctor Who
Clocked Your Mind

In 1952, a physician turned information theorist wired ten lamps into a rough circle, one Morse key waiting under each finger, and lit them in random order. He was not testing reflexes. He was pricing something no one had priced before: the cost, in bits, of making up your mind.

Part I

Same Test, Seventy-Four Years Later

W. E. Hick sat subjects in front of his lamp circle and had them press the matching key as fast as they could, varying only how many lamps might light. Below is the same test, digitized: a symbol appears, and you find its twin among a grid that changes size every trial.

Sixteen trials. The option count steps through 2, 4, 8, and 16, in an order you cannot predict. Tap the wrong symbol and it simply waits for the right one — that does not count against your time. Tap correctly and the next trial starts immediately.

Predict

Your next trial doubles its options — say, from 4 up to 8. Does your decision time double too?

Reaction board

16 trials · N steps through 2, 4, 8, 16, order randomized

A symbol appears. Find it among the options below and tap it — as fast as you can.

The option count changes every trial: 2, 4, 8, or 16, in random order.

Run the board above and this paragraph will fill in with your own fitted line, set beside Hick’s 1952 average of roughly five bits per second.

Part II

One Formula, Two Names

Hick’s own 1952 result, for n equally likely options, was T = b × log2(n + 1) — no intercept, just the log. A year later, Ray Hyman generalized it to RT = a + b × H, where H is Shannon entropy: the cost of any choice, weighted by how surprising each outcome actually is.

This page fits Hyman’s general form — with a real intercept a, because no human being has zero overhead before they start choosing — to Hick’s original equiprobable-n case. That is the form behind every number on this page: RT = a + b · log2(n + 1). Hick’s reduced version is the special case where a happens to be zero.

The +1 is already inside Hick’s own formula — it is not a second, rival convention the way Fitts’s Law (Week 27) has one. It prices the option of responding to nothing at all: in Hick’s apparatus, a lamp that has not lit yet is a real state of the world. Every interface choice includes that same silent option — close the tab, hit back, do nothing — whether or not you draw it.

The cost of a choice is not the count of options — it is the uncertainty of the outcome.

Hyman, 1953 — via the dossier

Hyman proved this by varying the frequency of each option and the predictability of one option given the last, independently of how many options there were — and got the same straight line in bits every time. Ten options where one is chosen 90% of the time price out near a two-option menu. A default, a memory of last time, a frequency-sorted list — each one lowers H without deleting a single option.

The law also has a fence around it. It prices deciding among a known set with a known mapping — not searching an unfamiliar list (which costs time roughly linear in length, not logarithmic) and not comprehending unfamiliar options (which is a reading-time problem the law does not touch at all). And the slope itself is not fixed: high stimulus-response compatibility and sheer practice flatten it, sometimes toward zero — which is exactly why a dense expert tool can be fast for the person who lives in it and paralyzing for everyone else.

Part III

Subdividing Is Not Free

The standard advice is to subdivide a long list into steps rather than delete from it. That is usually right — but not because the raw arithmetic gets cheaper.

Take the same sixteen options from the board above. Show them on one screen, or split them into four steps of four. Each screen you add pays its own intercept a and its own +1 — the leave-alternative gets priced again at every step. Using your own fitted line (or Hick’s 1952 reference if you have not run the board), here is what each shape actually predicts.

One menu, two shapes

using Hick's 1952 reference (run the board above for your own)

One screen, 16 options817 ms

Four steps of 4 options1858 ms

One screen · n=16: 817 ms; a + b·log2(17) = 0 ms + b·4.09
4 steps · n=4 each: 1858 ms; 4·(a + b·log2(5))
Difference: +1040 ms; staging costs more, not less

Modelled, not measured — both bars are computed from RT = a + b · log2(n + 1), once per screen. This is not the dossier’s own worked arithmetic; it is this page’s extension of the dossier’s Rule 3 (“subdivide instead of amputating”), run as a direct check against your own numbers. The dossier’s real claim for subdivision is scannability and error recovery — a search-and-mistake benefit Hick’s arithmetic does not model at all — not a shorter decision time.

Part IV

Sequencing, Not Deletion

The strongest real application of this law is not a shorter menu. It is the same menu, staged.

A profile-setup wizard that asks one question per screen, each with a sane default pre-selected and a visible “Skip”, prices out far cheaper per moment than one long form with the same fifteen fields — even though total capability never shrank. Slack’s onboarding reveals channels and integrations only after the first message is sent. Checkout funnels remember your last address and payment method, collapsing a five-way choice into a one-tap confirmation for anyone who has been here before.

Jakob Nielsen’s progressive-disclosure guidance supplies the discipline underneath every one of these: show the few things most users need most of the time, defer the rest behind a clearly labeled door, and decide the split with usage data, not opinion. Get that split wrong and every step becomes a toll on your most common task, not a kindness to your rarest one.

Part V

Design The Menu Yourself

Group N items into k categories — a two-tier menu, category then item — and watch the modelled total move.

Too few categories and each one is still a big flat list. Too many, and you are paying the two-screen tax — the intercept and the +1, twice over — more times than the items actually warrant. Slide k and find where the curve turns back up.

Menu designer

using Hick's 1952 reference

Categories (k): 6; 4 items each
Two-tier total: 1026 ms; modelled: 2 screens, each its own +1
One flat list: 929 ms; all 24 on one screen
Best k for this n: 4; 1026 ms

Total items (n)24

Categories (k)6

Modelled: total = (a + b·log2(k + 1)) + (a + b·log2(⌈n/k⌉ + 1)), one intercept per screen. This composes the dossier’s Rule 3 (subdivide) with Rule 1 (price in bits, not items) into a single curve — the dossier states the rules separately; the U-shaped curve is this page’s synthesis of them, not a number the dossier itself reports.

Challenge · Twelve options

Open

A first screen currently dumps all twelve options at once. Split them into equal steps so the predicted decision time falls under the target — but no step may hold more than 4 options.

Split 12 options into

Step 1

Step 2

Step 3

Step 4

Step 5

Step 6

Step 7

Step 8

Step 9

Step 10

Step 11

Step 12

Steps: 12; 1 option each
Predicted total: 2400 ms; modelled — see Note below
Target: 1497 ms; not yet

Try the fewest steps the 4-option limit allows before you try more of them.

Insight

Deleting an option is the expensive way to buy speed. Making one option probable is the cheap way.

Hyman’s half of this law is the half almost every “Hick’s Law” article skips: the price of a screen is not its item count, it is its entropy. A cluttered menu where one choice already gets 90% of the taps is priced like a two-option menu, not a ten-option one — and a minimal, two-option menu with no clear default can cost more attention than it looks like it should. This is also exactly where the law stops and a different, more contested literature begins: Hick’s Law says a bigger assortment takes measurably longer to choose from. It says nothing about whether you enjoy choosing, finish choosing, or regret it after — that is the choice-overload literature, and its average effect across fifty studies is close to zero. Cite the one you can actually defend.

Mental model

Whenever a screen asks someone to decide among several things,

look for the entropy behind it — how predictable the outcome already is — not the raw item count,

because decision time tracks log2(alternatives + 1), and lowering the odds of surprise buys more speed than deleting an option ever will.

Tomorrowfind your product’s least-used menu of eight or more items and check whether one option already gets most of the taps. If it does, that is your next default — before you delete anything.

End of experience

← Back to the constellation

The Doctor WhoClocked Your Mind

Same Test, Seventy-Four Years Later

One Formula, Two Names

Subdividing Is Not Free

Sequencing, Not Deletion

Design The Menu Yourself

Where this idea goes next

The Doctor WhoClocked Your Mind

Same Test, Seventy-Four Years Later

One Formula, Two Names

Subdividing Is Not Free

Sequencing, Not Deletion

Design The Menu Yourself

The Doctor Who
Clocked Your Mind

The Doctor Who
Clocked Your Mind