Algebra grids: every row is an equation, and the shapes are the unknowns

0:00Here is a puzzle that looks like a game and is not. A grid. Each row holds some drawn shapes, and at the end of the row there is a number. The rule is that the number is the total of the shapes beside it. Three squares, and twenty-seven. Two circles and a square, and nineteen. That is all you are told. And it is enough to work out exactly what a square is worth, and exactly what a circle is worth.

0:30The interesting question is not what the answer is. It is why a picture can settle anything at all. Because on the face of it you have two facts and two things you do not know, and that ought to be nowhere near enough. Start by taking the pictures away. Write a letter inside every square and a different letter inside every circle. Now the first row says s plus s plus s is twenty-seven, and the second says c plus c plus s is nineteen.

1:05Nothing was lost doing that. Which tells you something: the shapes were never doing any mathematical work. Nothing about squareness, nothing about the colour. They are names. But there is an assumption hiding in there, and it is the assumption the whole puzzle rests on. A square means the same number in every row. Every square, everywhere on the grid, is the same number. Drop that, and no row would tell you anything about any other row, and there would be nothing to solve.

1:41Now, where do you begin? Look at the first row. Every cell in it holds the same shape. That is the row that gives itself away, because it has only one unknown in it. Three of one thing, adding to twenty-seven. The second row cannot do that. It has two unknowns in it, and one statement about two unknowns pins down neither. So if a grid has a row made entirely of one shape, that is where you start. Always.

2:13And you do not have to work the rows in the order they are drawn. Start wherever the arithmetic is easiest. Three squares make twenty-seven. So a square is nine. Take a second on why that step is allowed, because it goes past very fast. Twenty-seven has been split into three equal parts, and each part is one square. That IS what dividing means. Not a rule you apply - the thing the row is already saying.

2:46And now write nine into every square on the grid at once. Not just the row it came from. Every one. That is the assumption from a minute ago finally earning its keep. So look at the second row again. It was two circles and a square, totalling nineteen. The square is not unknown any more. It is nine. Two circles and nine make nineteen. And notice what just happened to that row. It had two unknowns in it and now it has one. It has become the kind of row we already know how to finish.

3:24That is the whole engine of this thing. Find a row with one unknown, solve it, and watch another row lose an unknown. Two circles and nine make nineteen. Collect the circles first. There are two of them, so that is twice the circle, plus nine, equal to nineteen. That collecting step is the one most people skip, and it is the one that turns a picture into something you can operate on.

3:55Take nine off both sides: twice the circle is ten. Share both sides by two: the circle is five. Nine and five. Two shapes, two rows, and no guessing anywhere. Now check it, and check it against both rows, not just the one you finished on. First row: nine and nine and nine. Twenty-seven. Right. Second row: five and five and nine. Nineteen. Right. This matters more than it sounds. Two values that satisfy the row you solved last are guaranteed - you built them from it. The question is always whether they survive the other rows.

4:39These do. So they are the answer, and not merely an answer. Now a harder one, and the difficulty is not the arithmetic. Three rows this time. Two squares and a circle make twenty-seven. Two circles and a square make twenty-one. And a third row whose total has been left blank. Look for the row made of one shape. There isn't one. Every row has both unknowns in it. There is nowhere to start.

5:11So the move that worked last time is simply not available, and this is exactly the moment most people stall. Here is what to do instead. If no row has one unknown, MAKE one. Take the first row twice. Two of it: four squares and two circles, totalling fifty-four. Now take the second row away from that. The two circles cancel with the two circles, and one square comes off the four.

5:42What is left is three squares making thirty-three. A row with one unknown, which we built rather than found. So the square is eleven. Put that into the second row: two circles and eleven make twenty-one, so twice the circle is ten, and the circle is five again. Check both rows. Twenty-two and five is twenty-seven. Ten and eleven is twenty-one. Both hold. Which leaves the third row, the one with the blank total.

6:16People treat that blank as a third thing to find, and then complain there are too many unknowns. It is not. The moment the shapes are known, that total is fixed - there was never anything to choose. Five, eleven and five. Twenty-one. But look more carefully. That third row holds a circle, a square and a circle. The second row held a circle, a circle and a square. Same three shapes. Different order. And adding does not care about order.

6:50So the third row's total is twenty-one, and you can say so without doing a single sum. That is not laziness. That is seeing what the row actually says. One more grid, because it contains something genuinely strange. A circle and two diamonds make eighteen. A diamond and two circles make fifteen. Same move as before: take the second row twice, take the first away, and three circles make twelve.

7:23So the circle is four and the diamond is seven. Check: four and fourteen is eighteen; eight and seven is fifteen. The third row repeats the second exactly, so its total is fifteen with no work at all. And then the fourth row. Every cell of it is empty, the total included. You cannot complete that row, and not because it is hard. Three empty cells could hold twelve, or fifteen, or eighteen, or twenty-one, depending on what you put in them. A row constrains its shapes only if it has shapes in it.

8:02So step back and look at what we were actually doing. Each row was an equation. The grid was several equations about the same unknowns, all true together. Solving it was three moves and no others: find a row with one unknown, or build one by combining rows; solve it; and carry the answer into every other row. The shapes never mattered. Neither did the picture. What mattered was that a symbol held one fixed number everywhere it appeared, which is the entire reason the rows could talk to each other.

8:41And that is worth knowing, because guessing would also have found nine and five. It would not have found eleven and five, and it will not find anything on a grid with five shapes in it. So the next time a puzzle hands you shapes and totals, do not look for a clever guess. Write the rows out and see which one has a single unknown standing in it. The picture was never the point. It was equations, drawn.