Whack-a-Mole
Four times the holes is only about a quarter harder. Fitts's law explains why, and the clock is what actually decides the game.
Four times the holes is not four times harder
A 6 × 6 board has 4 times as many holes as a 3 × 3, so it ought to be about 4 times harder. It is nothing like that. The board is the same 420 pixels across whichever grid you pick, so your hand travels almost exactly the same distance either way — the mean hop between two holes varies by only 3.2 per cent across every grid here. All that really changes is how big the targets are, and the cost of a smaller target is logarithmic rather than linear.
Put through Fitts's law — the standard model for how long it takes to point at something — 4 times the holes works out at about 24 per cent more movement time, not 300 per cent. The table below has the figures.
What actually decides the game is the clock. The window each mole stays up shrinks as you hit them, and once it drops near your movement time there is no budget left to react in — on any grid, including the small one. That is why the window is shown live above the board, and why each grid stops shrinking at its own floor rather than a shared one: a 6 × 6 held to the 3 × 3's floor would be unwinnable by construction, and you would have no way to tell that it was the game rather than your hand.
What each grid actually costs
Fitts's law, Shannon form: ID = log₂(2D/W + 1) bits, movement time = 0.15 + 0.15 × ID seconds. Distances are the mean over every pair of holes, computed rather than estimated.
| Grid | Holes | Hole width | Mean hop | Difficulty | Movement time | Fastest window |
|---|---|---|---|---|---|---|
| 3 × 3 | 9 | 109 px | 229 px | 2.38 bits | 506 ms | 626 ms |
| 4 × 4 | 16 | 82 px | 225 px | 2.70 bits | 555 ms | 675 ms |
| 5 × 5 | 25 | 66 px | 223 px | 2.96 bits | 595 ms | 715 ms |
| 6 × 6 | 36 | 55 px | 222 px | 3.19 bits | 628 ms | 748 ms |
The mean hop barely moves because it is converging on a known constant: the average distance between two random points in a square is 0.5214 times its side, which on this board is 219 pixels. A finer grid just samples that square more densely. The constants in the movement-time formula are the usual mouse-pointing values from the human-computer-interaction literature; they differ between studies and between people, so treat them as comparing the grids with each other rather than as predicting your own hand.
How to use
- Pick a grid and a round length, then press Start.
- Hit each mole before its window closes — a faster hit scores more.
- Watch the window figure: it shrinks with every hit you land.
- Try the same round on a bigger grid and see how little it changes.
Frequently asked questions
Is a bigger grid much harder?
Far less than it looks. A 6 by 6 has four times the holes of a 3 by 3, but the board stays the same size, so your hand travels almost the same distance either way — the mean hop between holes varies by about three per cent across every grid here. Only the targets get smaller, and by Fitts's law that costs roughly a quarter more movement time rather than four times.
What is Fitts's law?
A model of how long it takes to point at something: movement time rises with the distance to the target and falls with its size, logarithmically rather than linearly. In the Shannon form used here, the index of difficulty is log base two of (2D/W + 1) bits, and movement time is a fixed overhead plus a cost per bit. It has held up across mice, touchscreens, styluses and actual limbs since 1954.
Why is the distance almost the same on every grid?
Because it is converging on a constant. The average distance between two random points in a square is 0.5214 times its side, which is a known closed form rather than an estimate. A finer grid just samples that same square more densely, so the mean hop settles rather than growing — which is exactly why more holes does not mean much more work.
What actually makes it hard, then?
The window. Each mole stays up for a shorter time as you land hits, and once that window drops near your movement time there is nothing left to react in. Halving the window removes the entire budget on any grid, including the easiest — something no change of grid size ever does. That is why the live window is shown above the board.
Why does each grid stop at a different speed?
Because the floor is derived from that grid rather than picked. Every grid stops shrinking at its own predicted movement time plus a fixed slice of slack, so a big grid is hard without being impossible. A 6 by 6 held to a 3 by 3's floor would be unwinnable by construction, and you would have no way to tell that it was the game rather than your hand.
How is the score calculated?
A hit early in the window is worth more than one at the end, up to eleven points and never less than one. The game is about the window, so scoring on where in the window you landed measures the thing being played rather than just counting hits — but a slow hit still beats a miss, which is why the floor is one rather than zero.
Are mouse and touch scores ranked together?
No, they are separate boards. This is aiming, which is precisely the case where the input is the skill — a mouse and a thumb are not doing the same physical task, and one shared table would mostly rank who was using which. Input is read from the events that actually played the round rather than guessed from your browser.
What is a good score?
Depends entirely on the grid and the round length, which is why each combination has its own board. Within one setting, accuracy matters more than most people expect — a run that hits ninety per cent of the moles on a slow window usually beats a frantic one that misses a third of them, because a miss costs the whole window rather than a few points.
Does the mole ever appear twice in the same hole?
No, never immediately. A mole popping straight back up where it just was reads as a bug, and it also makes that move free — your hand is already there — which would break the model the difficulty is built on. The next hole is always a different one.
Is this a real reaction test?
Not quite. A reaction test measures the time between a signal and a response with no movement involved; this measures movement as well, which is why Fitts's law applies at all. If you want the pure figure, use a dedicated reaction time test — the number here is reaction plus travel, and travel is usually the larger part.
Do the movement time figures apply to me?
Treat them as comparing the grids with each other, not as predicting your own hand. The constants are the standard mouse-pointing values from the literature, and they vary between studies, between devices and between people. The relationship between the grids holds regardless; the absolute milliseconds are a guide.
Does this store anything?
Only if you post a score. The game runs entirely in your browser and nothing is saved between rounds unless you choose to put a score on the global board, which sends the name you type and the number.
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