The Moves: How to Actually Get an Idea

Phase 1 gave you the loop. But the loop has a hole in it the size of a house: step 2 says "devise a plan," and you reasonably want to scream, with what? Where does a plan come from when you're staring at a blank page?

It comes from a short menu of moves. These are the windows a mathematician walks around the building trying. None of them is guaranteed to work. All of them are worth a try, and on most problems at least one of them cracks the door. Learn the menu and "I have no idea" turns into "let me try the first three."

Move 1: Try the smallest case

When a problem is stated for some big or general number, shrink it. Do the case for 1. Then 2. Then 3. Small cases are cheap, and they show you the machinery.

Problem: What is 1 + 2 + 3 + ... + 100?

Shrink it:
  n = 1:  1                 = 1
  n = 2:  1 + 2             = 3
  n = 3:  1 + 2 + 3         = 6
  n = 4:  1 + 2 + 3 + 4     = 10

What just happened: you stopped trying to leap at 100 and did the parts you can do in your head. Now you have data — 1, 3, 6, 10 — and data is something to look at, which a blank page never was.

Move 2: Find a pattern

Once you have small cases, stare at the numbers and ask what they're doing. Patterns are the bridge from "examples" to "rule."

Sums:    1,  3,  6,  10
n:       1,  2,  3,  4

Try multiplying n by the next number:
  1×2 = 2,  2×3 = 6,  3×4 = 12,  4×5 = 20
That's exactly double each sum. So sum = n(n+1)/2.

Check at n = 100:  100 × 101 / 2 = 5050.

What just happened: the pattern n(n+1)/2 didn't fall from the sky — you found it by poking the small cases until the numbers confessed. You also checked it on a fresh case before trusting it, which is the look-back habit from phase 1 doing its job.

Move 3: Work backwards

Some problems are a maze. Walking forward from the start, every turn looks equally plausible. So start at the exit and walk back — the goal often has only one thing that could lead to it.

Goal: end holding exactly 4 liters, with a 5-liter jug and a 3-liter jug.

Work backwards: to have 4 in the 5-jug, I'd pour 1 out of a full 5.
  To pour exactly 1 out, the 3-jug must already hold 2 (it has room for 1).
  To have 2 in the 3-jug... fill 5, pour into 3 (leaves 2 in the 5),
  empty the 3, pour those 2 in. Now the 3-jug holds 2.
Read it forwards and you have the solution.

What just happened: forwards, the jug puzzle has a dozen pointless moves you could make. Backwards, each step had almost no choice — "to get here, I must have come from there" pruned the maze down to a path.

Move 4: Solve a simpler version

If the real problem has three hard things tangled together, untangle one. Drop a constraint, lower a dimension, assume the nice case. Solve that, then add the hardness back.

Real: shortest route visiting 12 cities.
Simpler: shortest route visiting 3 cities — trivial, only a few orders.
         Then 4. You start to see why it explodes, and which ideas
         (nearest-neighbor, swapping two stops) even make sense to try.

What just happened: the simpler version didn't solve the 12-city problem, but it taught you the shape of it and which approaches are worth scaling up — far better than guessing at the full thing cold.

Move 5: Look for an invariant or symmetry

This is the deepest move, so it gets its own home in phase 3 — but meet it now. An invariant is something that doesn't change no matter what moves you make. If you can find one, it often settles the whole problem in a line.

Tile a chessboard with dominoes, each covering one black + one white square.
Now remove two opposite corners — both the same color, say both white.

Invariant: every domino covers exactly one white and one black, always.
So any tiling covers equal whites and blacks.
But the mutilated board has 32 black and 30 white. Unequal → impossible.

What just happened: you didn't try a single tiling. The invariant ("one white, one black, every time") made all tilings answer the question at once. That's the power — one unchanging fact replacing infinite case-checking.

For builders

These map straight onto how you cut down a bug. "Smallest case" is the minimal reproduction. "Solve a simpler version" is commenting out half the system to localize the fault. "Work backwards" is reading a stack trace from the crash up to the cause. "Find an invariant" is the assertion that should always hold — and the moment it doesn't, you've found your bug.

[
  {
    "q": "You face '1 + 2 + ... + 1000 = ?' with no formula in mind. What's the natural first move?",
    "choices": ["Add all 1000 numbers carefully", "Try the smallest cases (n=1,2,3) and look for a pattern", "Give up and look it up", "Guess a round number"],
    "answer": 1,
    "explain": "Small cases turn a blank page into data (1, 3, 6, 10...), and the pattern n(n+1)/2 then jumps out."
  },
  {
    "q": "What makes 'work backwards' powerful on a maze-like puzzle?",
    "choices": ["It's faster to write", "The goal usually has only one thing that could lead to it, pruning choices", "It avoids arithmetic", "It always finds the shortest path"],
    "answer": 1,
    "explain": "Forwards every move looks plausible; backwards each step asks 'what must I have come from?', which often has a near-unique answer."
  },
  {
    "q": "Why does the invariant settle the mutilated-chessboard problem without trying any tiling?",
    "choices": ["Because the board is small", "Because every domino always covers one black and one white, so all tilings need equal counts — which the board lacks", "Because corners don't matter", "Because dominoes are symmetric"],
    "answer": 1,
    "explain": "One unchanging fact about every domino applies to every possible tiling at once, replacing infinite case-checking with a single contradiction."
  }
]

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