From Local to Global

In one sentence

Some quantities look messy piece by piece but neat as a whole. The two most useful moves:

Combine: add up (or multiply) all the local pieces and look only at the whole.
Count twice: work out the same whole in two different ways. The two results must be equal. If they are not, you have a contradiction; if they are, you have an equation.

Predict first

Fill each cell of a $3 \times 3$ grid with $+ 1$ or $- 1$ . Can every row multiply to $- 1$ while every column multiplies to $+ 1$ ?

YesNo

Choose an answer first; then you can see the result.

Example: all row products −1, all column products +1?

Step 1 / 4

Locally: fix the rows, break the columns
Making all three row products $- 1$ is easy. But then all three column products are $- 1$ too. Change one more cell and a row breaks again.
Globally: the product of all nine, by rows
The same product, by columns
Compare the two

Stop 1 · Counting pieces by rows and by columns (Vol. 7)

Counting twice can tell you it cannot be done, and also how many there must be when it can.

Example: a 4 × 6 board with 3 pieces in each row and 2 in each column?

Step 1 / 4

Count by rows
$3$ pieces in each of $4$ rows: the total should be $3 \times 4 = 12$ .

$3 \times 4 = 12$
Count again by columns
Agreeing is not enough: build it
Change the goal and it fails

Stop 2 · Each move changes only a small part (Vol. 7)

In problems about operations, every move is “local”. To decide whether you can ever succeed, look for a global quantity that no move changes.

Example: flipping whole rows and columns: can everything become +1?

Step 1 / 4

The rules
In a $4 \times 4$ grid only the top-left cell is $- 1$ . Each move flips the signs of all four numbers in a whole row or a whole column. Can you make every number $+ 1$ ?
Try one move
Find a global quantity that does not change
Conclusion

Stop 3 · Treat a chunk of the expression as one whole (Vol. 1)

Algebra works the same way: do not rush to expand every bracket. First look for one whole block that keeps coming back.

Example: factor (x+1)(x+2)(x+3)(x+4) − 24

Step 1 / 4

Do not expand everything
Multiplying out all four brackets gives a messy quartic. Pair them up first, so that the two products look as alike as possible. $1 + 4 = 2 + 3$ , so pair the outer two and the inner two.

$(x + 1) (x + 4) \cdot (x + 2) (x + 3) - 24$
The same block appears
Treat it as one whole
Substitute back

When evaluating, too, you often do not need each letter, only the “whole”: if $a + b = 5$ and $ab = 3$ , then $a^{2} + b^{2} = (a + b)^{2} - 2 ab = 25 - 6 = 19$ . There is no need to solve for $a$ and $b$ at all.

Practice

Exercise 1Basic

$10$ students join after-school clubs. Each student joins exactly $3$ clubs, and each club has exactly $5$ students. How many clubs are there?

Exercise 2Advanced

Given $x^{2} - 3 x + 1 = 0$ , find the value of $x^{2} + \frac{1}{x ^{2}}$ .

Exercise 3Challenge

(Adapted from a classic problem) Fill each cell of a $5 \times 5$ grid with $+ 1$ or $- 1$ . Work out the $5$ row products and the $5$ column products, and add these $10$ numbers together. Prove that the sum can never be $0$ .

In one sentence

Example: all row products −1, all column products +1?

Stop 1 · Counting pieces by rows and by columns (Vol. 7)

Example: a 4 × 6 board with 3 pieces in each row and 2 in each column?

Stop 2 · Each move changes only a small part (Vol. 7)

Example: flipping whole rows and columns: can everything become +1?

Stop 3 · Treat a chunk of the expression as one whole (Vol. 1)

Example: factor (x+1)(x+2)(x+3)(x+4) − 24

Practice

Mistake Museum · Exhibits from this stop

Counting twice, but not counting the same thing

Stops on this line