In one sentence
Many problems do not tell you what the answer is. So work it out yourself:
try a few small cases → spot the pattern → write it as a conjecture → prove it.
Guessing gives you a direction; proving is what makes it an answer. Guess without proving and you can crash; try to prove without guessing and you do not know where to start. You need both steps.
Predict first
What is ? Do not just add it up: look at the table above and make a guess.
This is the sum of the first odd numbers, and it equals . Why do odd numbers always add up to a perfect square? Follow it step by step below.
Example: the sum of the first n odd numbers
Step 1 / 4- Work out small cases
; ; . Do them by hand, do not skip.
Stop 1 · Guessing a linear function from a table (Vol. 3)
Make a row of joined squares out of matchsticks. How many matches do you need for squares?
Example: how many matches for n squares?
Step 1 / 4- Count small cases
square takes matches, squares take , squares take .
Stop 2 · Guess a root, then divide (Vol. 1)
A cubic has no ready-made cross method. But you can guess a root first.
Predict first
. Which of these numbers is a root?
and , so neither works. : that guess is right.
Example: factor x³ − 4x² + x + 6
Step 1 / 4- Narrow down the candidates
If an integer is a root, then , so must divide . The only candidates are .
Stop 3 · A pattern in a table of powers (Vol. 6)
The first guess at this stop will be wrong. Once it is corrected, it becomes a famous theorem.
Example: the remainder of n to the power m − 1 when divided by m
Step 1 / 5- Mod 5: look at the n⁴ column
Apart from row , the column is all s.
Stop 4 · Taking stones (Vol. 7)
Game problems are perfect for guess-then-prove: you can play the small positions by hand.
Example: take 1 to 3 stones each turn; whoever takes the last stone wins
Step 1 / 4- The smallest piles
, or stones: the first player takes them all and wins. stones: however many the first player takes, the to left are all taken by the opponent, so the first player loses.
If you guess, you must prove
Predict first
Put points on a circle and join every pair with a chord (no three chords meet at one point inside the circle). For the circle is cut into regions. How many regions are there for ?
There are regions. Every chord and every crossing point inside the circle cuts off one more region: regions chords crossings. With points there are chords and crossings, so .
Here is an even nastier one: the expression below gives primes for dozens of terms in a row.
Practice
A convex quadrilateral has diagonals, a convex pentagon has , and a convex hexagon has . First guess a formula for the number of diagonals of a convex -gon and explain why it holds. Then answer: how many diagonals does a convex 12-gon have?
Let . Work out , guess a formula for and prove it, then give . (Write the answer as a fraction, such as 3/4.)
A pile has stones. Two players take turns; on each turn you may take , , or stones, and whoever takes the last stone wins. If the first player wants a guaranteed win, how many stones should they take on the first move?