Stained Glass

Skip-counting turned into a picture: the pattern each times table draws across a hundred square.

Grades: 1–6
Subjects: Counting & Estimating, Adding & Subtracting, Multiplying & Dividing
Kind: Puzzle
Cost: Free, no account, no ads

What it is

The board is a hundred square, 0 to 99, with every number face down and one line of instruction above it: skip count, 0, 3, 6, 9, and then it stops. Turn over a number that belongs to the count and it paints the patch of canvas nearest to it. Turn over one that does not and the pane arrives in a second colour scheme that looks nothing like the first, which is the board telling you so without a word.

What it gives you to practise

Skip counting, and then something the times tables are rarely asked: what shape a count makes. The fives are two columns down a hundred square, the nines walk a diagonal one square left each row, and the last board in the pack lets you drag the right-hand edge from five squares wide to fifteen and watch the same hundred numbers rearrange into a different pattern. A pattern on a hundred square turns out to be a pattern of the width as much as of the count. Every board also begins at 0 rather than 1, because zero threes make zero — so zero belongs to every count, and the first tap of every board is the top-left square.

In a classroom

The free board at the foot of the pack has no rule to break and nothing to finish: it is a hundred square and a handful of colours. It is the one worth a projector. Ask a class which width will turn the fives into a single column before anybody drags the edge, then let the board settle the argument.

The rules

Tap the gray dots to reveal the numbers underneath. Each revealed number paints the part of the canvas closest to it (its Voronoi region), and the colour tells you whether that number belongs in the answer. The result is a window of stained glass that grows as you choose numbers.

How to play

Read the big line above the board. It starts the count for you: "Skip count: 0, 3, 6, 9, …" — the first four numbers, which is enough to say what the step is. Work out the step and keep going.
Tap a gray circle to reveal the number under it. The glass instantly grows a new coloured pane around that number.
Tap a revealed number again to hide it (its region disappears).
When only the numbers that match the rule are up, the board sees it for itself. Half a second later — once the last pane of glass has grown in — the Done button turns green where it stands. Press it again and it takes you back to the grid of windows.

You never have to hand anything in. The board is watching, and the button going green is the whole announcement. But you can press Done whenever you think you have finished. If you have, it ends the board there and then. If you have not, it does what Help does: it points at the first square that is wrong, so pressing it always tells you something.

The grid you came from holds every board in the pack, and the window you just painted is now sitting in its place there. Pick the next one yourself: along a row is the same count on a bigger board, down a column is a new count on the same size.

Stuck? Press Help

Help plays the tap you have not made yet. One square switches on and off again — once, slowly — and its pane of glass swells and is pulled back with it. The pane never arrives, so nothing is decided for you.

A face-down number growing a pane that keeps being taken away — that one belongs in the count. Turn it up and the pane stays.
A number that is face up having its pane pulled back and returned — that piece of the window should not be there. Turn it down.

Which of the two you get tells you what kind of mistake it is; finding out why is still yours. It always takes the first mistake, reading the board the way the numbers run, so pressing it again and again walks you along your own board in order. On a finished board it does nothing — there is nothing left to point at.

Painting

You don't have to tap them one at a time. Hold down and drag — with a finger or with the mouse — and every number the stroke passes over turns up together.

There is clear space between the dots, so a stroke only paints what it actually goes through. That means you can draw the pattern: the multiples of 5 are two straight columns down a hundred square, the multiples of 9 are a diagonal, the even numbers are five columns. Rule the line and the whole set turns over at once.

If a stroke starts on a number that is already face up, it turns numbers down instead — so drawing back along a line you got wrong rubs it out.

Every board starts at zero

The first square is 0, not 1. A ten-by-ten board holds 0 to 99, a five-by-five holds 0 to 24.

That matters for skip counting, because zero is a multiple of every number. Zero threes make zero, so 0 goes in the threes. Zero sevens make zero, so 0 goes in the sevens too. That is why the line above the board always begins at 0 — and why every count on every board starts by tapping the top-left square.

How the colours work

Every board is painted from two colour schemes, and which one a number gets is the board's answer to you:

Numbers that do match the rule are painted from the first scheme — say, bright pastels.
Numbers that do not match are painted from a second scheme that looks nothing like it — say, deep inks.

So turning a number over tells you whether it belongs. Use it to check yourself: if you tapped 27 looking for multiples of 4 and it comes up in the wrong colours, tap it again to put it back.

The matching numbers are painted in counting order along their scheme, so a finished skip-count is a gradient running across the board — 0 at one end of the colours, the last multiple at the other.

Press Clear and the board empties and picks two brand new schemes (Done loses its green with it — an empty board is not a finished one). The same puzzle paints a different window every time you play it, which is the reason to play it again.

The board

Numbers run row by row — 0 in the top-left, then left-to-right, top-to-bottom, just like a hundred square.

Rows (5×5)
 0  1  2  3  4
 5  6  7  8  9
10 11 12 13 14
15 16 17 18 19
20 21 22 23 24

That layout is what makes the counts draw themselves. Every tenth number is a column; the nines walk down a diagonal one square left each row; the fives are two columns. Once you have seen the shape, you can paint it in one stroke.

Free exploration

The last board in the pack, under the counts, is headed by a 1 — a hundred square counted by ones. Every number belongs to that count, so nothing you turn up can be wrong: there is no rule to find and no mistake to make. Paint the whole window, or stop wherever it looks best.

That board has no Help button, for the same reason: Help points at the first square that disagrees with the rule, and nothing there can disagree. It has no Done either, and nothing there ever turns green — there is nothing to finish. Leave when you like: the window you painted is waiting in its place at the foot of the grid.

And because nothing can be wrong there, the colours have no answer left to give — so that board hands them to you. Under it sits a row of seven round chips. There is no writing on any of them: each one is a picture of what it does.

The stripe is the whole set of colours, in counting order. Pick it and the board colours by number the way it always does — 0 at one end of the colours, 99 at the other. This is the one you start with.
The wheel is the five colours mixed. Pick it and every square you paint takes one of them at random.
The other five chips are the colours themselves, five stops along the same stripe. Pick one and everything you paint comes out in it.

Drag with a colour in hand and the whole line comes out in it — including over squares already painted in some other colour, which you paint straight over. Drag back over squares you painted in the colour you are holding and they turn face-down again, exactly as everywhere else. So you can only rub out the colour you are holding: pick up the green and you cannot wipe the red by accident.

To put a square back to the board's own colour, turn it off and on again with the stripe picked.

Press Clear and you get a whole new set of five, cut from a whole new window.

Changing how wide the board is

That board also lets you move its right-hand edge. There is a grip on it, half way down, with an arrow pointing each way.

Drag it in or out and the board follows your finger.
Or tap it: the left arrow takes a number off every row, the right arrow adds one.

Either way the same hundred numbers spread themselves into rows of whatever width you stop at — anything from 5 to 15.

Nothing about the numbers changes. 0 is still 0 and 99 is still 99, and every square keeps the colour you painted it. All that changes is how many of them fit on a line — so the picture you painted rearranges itself in front of you:

Paint the tens on a ten-wide board and they are a single column down the left. Pull the edge in to five and they are still a column, every other row. Pull it to nine and they walk across the board one square right at a time.
Paint the fives on a ten-wide board and they are two columns. On a five-wide board they are one solid column and the whole rest of the paper is bare.
Anything counted by 3 makes columns whenever the width is 6, 9, 12 or 15 — a width the count divides — and diagonal stripes at every other width.

That is the thing worth playing with: a pattern on a hundred square is a pattern of the width as much as of the count. Paint a skip count, then drag the edge and watch what the same numbers turn into.

Packs

Skip Counting — a grid of boards: one row for each count from 2 to 10, and one board in the row for each size from 5×5 up to the full hundred square, with the free-exploration hundred square at the foot. Every number is shown from the start. Pick by looking — the pictures are the boards you have already painted.

Play Stained Glass