Number pyramids: what the top cell is made of

0:00One rule. Every cell holds the total of the two it rests on. That is the whole thing. Put one, nine and four along the bottom. One and nine make ten. Nine and four make thirteen. Ten and thirteen make twenty-three. Now look at the bottom row again. One, nine and four add to fourteen. But the top says twenty-three. So the top is NOT the bottom row added up, and the gap is nine.

0:30Hold on to that. It is the question this whole thing turns on. Going up is easy. Give me any bottom row and I can grind out every cell above it. Six and two gives eight. Three, four and three gives fourteen. Five, four, five and nothing gives thirty-two. Notice the last one. Its bottom row adds to fourteen, exactly like the first pyramid we built. But its top is thirty-two, not twenty-three.

1:03Same total along the bottom, different number on top. So whatever the top cell is made of, it is not just the total. But grinding upward is not the interesting direction. The interesting question is coming down. Here is a pyramid with almost nothing in it. Three rows. The top is ten. The left cell of the middle row is four. The bottom left is one. Three cells known, three cells blank.

1:36Because the rule works in both directions, subtraction opens it. Ten is four plus something, so the other middle cell is six. Four is one plus something, so the bottom middle is three. And six is three plus something, so the bottom right is three as well. Bottom row one, three, three. Middle row four, six. Top ten. Done, with nothing but taking away. Now try this one. Also three rows. Also three cells known.

2:11The top is sixty. The bottom left is twelve. The bottom right is eight. Everything else is blank. Where do you start? You cannot. Subtraction needs two cells of a triple already filled, and there is no triple here with two. Sixty sits above two blanks. Twelve sits beside a blank. Eight sits beside the same blank. Every move you would like to make is missing one number. Three cells known, and not one of them is any use. This is the moment worth stopping on, because what happens next is what algebra is actually FOR.

2:52And notice this is not a harder pyramid than the last one. It has just as much information in it. The information is simply in the wrong PLACES for subtracting. Put a letter in the blank bottom cell. Call it x. Not a solution - just a name. And now the rule does something remarkable. It manufactures the rest for you. The left cell of the middle row rests on twelve and x, so it is twelve plus x.

3:23The right cell rests on x and eight, so it is x plus eight. And those two together make sixty, because that is what the top cell means. Look at what has happened. x went in ONCE and came out TWICE, because the middle cell of the bottom row holds up both cells of the row above it. So: twelve plus x, plus x plus eight, is sixty. That is twenty, plus two lots of x, equal to sixty. Two lots of x are forty. So x is twenty.

3:58One letter. One equation. And the equation was not given to you - the rule made it. Now the rest walks out. Twelve and twenty make thirty-two. Twenty and eight make twenty-eight. And thirty-two and twenty-eight make sixty, which checks. Bottom row twelve, twenty, eight. That is the pyramid, and no amount of subtracting would ever have found it. And here is the thing worth noticing: naming ONE cell was enough, because the rule generated every other equation for free.

4:34This is not a one-off either. Four rows, apex fifty, two cells given along the bottom: the same move gives four, nine, six, one. Another one closes on five, fourteen, seven, two with seventy on top. A third closes on three, five, five, two. All three stall in exactly the same place, and all three need exactly the same move. Which is the point: this is not a trick for one pyramid, it is the method.

5:10Now back to the question from the beginning. What IS the top cell made of? Start as small as possible. Two cells at the bottom, one on top. Call them a and b. The top is a plus b. Each bottom cell counted once. No surprises. Three cells now. Call them a, b and c. The middle row is a plus b, and b plus c. So the top is a plus b, plus b plus c. Which is a, plus TWO b's, plus c.

5:44There it is. The middle cell is counted twice. And that is exactly the gap we started with. One, nine and four add to fourteen, and the top read twenty-three. The difference was nine - which is the middle cell, counted a second time. So the weights along a three-cell bottom row are one, two, one. Which raises the obvious question: for four cells, is it one, two, three, four?

6:13It is not. And the reason it is not is worth seeing properly, because guessing a pattern from one case is how people go wrong here. Here is where the twice actually comes from. Draw the pyramid as a network. From any cell, you can step up-left or up-right into the cell above. Now count the ROUTES from each bottom cell to the top. From the left corner there is exactly one route: up-right, up-right. From the right corner, one route. But from the middle, there are two - and that is the two.

6:50A cell's weight is not a mystery. It is how many ways up there are from it. Not a rule to memorise - something you can count. Which also tells you why the corners are always one. A corner cell has only one way to go at every step; there is no choice to make. Four cells along the bottom. Count the routes. The two corners have one each. And each of the middle two has three.

7:20One, three, three, one. Not one, two, three, four. So now you can find a top cell without building a pyramid at all. Four, thirteen and eight? Four, plus twice thirteen, plus eight. Thirty-eight. Seven, eleven and three gives thirty-two. Ten, fourteen and twenty-five gives sixty-three. And with four cells: eight, nineteen, twenty-one, thirteen. Eight, plus three nineteens, plus three twenty-ones, plus thirteen. A hundred and forty-one, straight off the bottom row.

7:56Build it out if you like. Row by row, cell by cell, all the way up. It comes to a hundred and forty-one. Seven, eighteen, nineteen, six gives a hundred and twenty-four. And nine, seven, five, eleven gives fifty-six. One last thing, and it is a lovely one. Take a sequence where every number is the total of the two before it: one, two, three, five, eight, thirteen, twenty-one. You may know it as the Fibonacci sequence; Virahanka described it centuries earlier.

8:35Lay the first three along the bottom of a pyramid. One, two, three. One and two make three. Two and three make five. Three and five make eight. Every single cell is in the sequence. Nothing ever leaves it. And that is not luck. Every cell above the bottom is two neighbours added - and in this sequence, two neighbours added is always the next term. The pyramid cannot escape.

9:05So which term ends up on top? Three rows gave eight, which is the fifth term. Try four rows: one, two, three, five along the bottom builds three, five, eight, then eight, thirteen, then twenty-one. Twenty-one is the seventh term. Five, then seven. Each row you add moves you two places along the sequence, not one - because each row is the sequence itself, shifted up two. So an n-row pyramid puts the term numbered two n minus one on top.

9:41Which means a twenty-nine-row pyramid does not hold the twenty-ninth term. It holds the fifty-seventh. And you can say that without drawing a single cell of it. That is the whole point. Once you know what the top is MADE of, you do not have to climb to it.