Finding an unknown by reasoning about what must balance

0:00Two pictures. Both balanced. Both hiding a weight from you. One of them you can crack with arithmetic you already have. The other cannot be cracked that way, and the reason is the whole video. The first: a bar hanging from a ring, with things on two strings below it. The second: a scale with a pan on each side, and the same unknown sack on both pans. That difference is the whole difference.

0:30Everything before it is counting, and everything after it needs a permission nobody has given you yet. Start with the hanging bar. It hands you two facts at once. The ring above it names the whole weight hanging below. And the bar hanging level says the two sides agree. Put those together. If the ring says twenty four and the bar is level, each side carries twelve. But be careful about why: it is not because there is a ring, it is because the bar is level.

1:05Here is a ring marked seven, with four on one string and three on the other. The ring is telling the truth: four and three really is seven. And this bar tips, so there are no halves to take. Take a real one. The ring says twenty four. On the left string: a starfish, a striped fish, a starfish. On the right: a striped fish and a grey fish. One number is given to you. A starfish weighs two.

1:36Halve the total first. Each side carries twelve. Now work inward on the left. Two and two is four, so the striped fish makes up the rest. It is eight. The right string also carries twelve, and eight of that is the striped fish. So the grey fish is four. Notice the shape: one unknown had to be found before the other could be. Here is one with a twist. The ring says eight.

2:06On the left, a brick. On the right, a shorter bar hanging from the first, carrying two identical cards. Halve the eight. Each side carries four. So the brick weighs four. And now look at the little bar. It is not decoration. It is a hanger in its own right, carrying four, and hanging level. Which means its two sides agree as well, and each card is two. A branch of a picture can be a whole claim by itself, solved the same way one level down.

2:41Two more, the same reasoning entered from opposite ends. Ring eighteen. On the left, a sun and four clouds. On the right, three lightning bolts. The sun weighs five. Each side carries nine. Five of the left is the sun, so four clouds share the other four. Each cloud is one. The right side then falls out: nine shared by three bolts, so each bolt is three. Now the reversal. Ring forty, and nothing is given to you at all.

3:14On the left, a crown and five gems. On the right, four crowns. Each side carries twenty. Nothing to start with on the left: both things on it are unknown. But the right side is four crowns making twenty, so a crown is five. Carry it back. A crown and five gems make twenty, so each gem is three. Now take the ring away entirely. A bar held level in somebody's hand. No ring. No total anywhere.

3:47On one side, a slice of watermelon. On the other, a banana, an orange and a second banana. You are told two things: the watermelon is ten, the orange is four. The bar is level, so the two sides agree. Ten on the left means ten on the right. Four of that ten is the orange, and the two bananas share what is left. Each banana is three. No ring, no halving, and still forced.

4:19One more of those, and something new happens. Four stars hang on the left. On the right, a ring, a star, and a second ring. A star weighs four. Read it straight off and it works. Four stars is sixteen, so the right side is sixteen. The star there takes four, and twelve is shared by two rings. Each ring is six. But there is a shorter way, and it is the one to watch.

4:48There is a star on each side. Take one star off the left and one off the right, together. The bar does not move, because you took the same weight off both sides. And what is left is smaller. Three stars against two rings. Twelve against two rings, so each ring is six. Same answer, different move. Hold on to that move. Now the picture changes: a scale with a pan on each side.

5:18Same idea, standing on the ground instead of hanging. Left pan: one sack and a two kilogram weight. Right pan: a ten and a two. Nobody says what the sack weighs. That is what you want. There is a two kilogram weight on each pan. So lift them both off at the same instant. Watch the beam. It does not move. Now the left pan holds only the sack, and the right holds ten kilograms.

5:48The sack is ten. Here is the one that breaks that method. Left pan: two sacks. Right pan: a ten, a four, and one more sack. Try halving. There is no total to halve. Try working inside the left pan. It holds nothing but sacks, and you do not know what a sack weighs. The trouble is that the thing you are chasing stands on both pans at once. So use the move from the stars. There is a sack on each pan.

6:20Take one sack off the left and one off the right, together. The beam does not move. You took the same weight off both sides, even though you do not know what that weight was. And now one sack sits against fourteen kilograms. The sack is fourteen. Do it again, harder. Left pan: five sacks. Right pan: two ten kilogram weights, a one, and two more sacks. Twenty one kilograms and two sacks, against five sacks.

6:53This time aim at something specific: get all the sacks onto one pan. There are two sacks on the right, and certainly two among the five on the left. So take two sacks off each pan. Three sacks left on one side, twenty one kilograms on the other. Three sacks make twenty one, so each sack is seven. And the one that looks frightening and is not. Left pan: ninety sacks and a fifty kilogram block. Right pan: sixty sacks and a five hundred kilogram block.

7:29Nothing new is being asked; only the numbers changed. There are sixty sacks on each pan. Take sixty off each. There is fifty kilograms on each pan. Take fifty off each. What is left is thirty sacks against four hundred and fifty kilograms. Thirty sacks make four hundred and fifty, so a sack is fifteen. Two moves — exactly the two you used on the small one. The size of the numbers changed nothing about the method.

8:02One habit to kill before we finish. Somebody looks at the two sack puzzle and says: I think it is fourteen. Two fourteens is twenty eight. Ten and four and fourteen is twenty eight too. It works. That check is true, and it is not the same thing as an argument. It tells you fourteen works. It tells you nothing about fifteen. To rule fifteen out you have to try it. Then sixteen. Then keep going.

8:32Taking a sack off each pan does something different. It does not test one answer. It rules out every other one, in a single move. Guessing that happens to land is not the same as reasoning that forces. So: where this leaves you, and where it stops. Everything in this video was decided by two moves and only two. Take equal weights off both sides, and the balance holds. Then split what is left into equal parts.

9:05That is a real method. Not guessing, and it worked on numbers of every size. But look at what did the work: a picture of a beam that does not move. The move is right because the scale would tip if it were not, and you can see that it does not. Next you make the same move with no scale in front of you. No beam, no pans, nothing to look at.

9:31Just the claim that two things are the same size, and an argument that you may do the same to both. Because a picture can only show what you can draw, and a few hundred of anything on one arm stops being an argument.