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Advanced Problem Restatement

when pushing harder stops working

Some problems don't give way to more effort, because the way you're looking at them has no answer in it. This is the system for those: a signal that tells you to stop pushing, a line that pins down what solved means, levers that change how you see the problem, and a check that the new version is still the problem you had.

  • Challenging to start
  • Life-Changing impact
  • Learn it in 30 days
  • A protocol for being stuck
same board, seen two ways
The mutilated checkerboard drawn plain and coloured, and the five stations of a stuck protocol A chessboard with two opposite corners cut off has 62 squares. Can 31 dominoes cover it? Drawn plain, an attempt places 30 dominoes and leaves 2 squares that don't touch, and searching other tilings has no end. Drawn coloured, the two cut corners are the same colour, leaving 30 dark and 32 light squares. Every domino covers one dark and one light, so 31 can't fit. The answer is no. Below, the protocol: 1, stuck signal, the same move three times with nothing new. 2, done when: a yes with a tiling, or a no with a reason. 3, change how you see it, with six levers: draw it, what never changes, try the extreme, work backward, change the unit, split who wants what; the one used here is what never changes. 4, drift check: keep, tighten or go back; this version was kept, because a no with a reason passes the done-when line. 5, solve and log the lever. The mutilated checkerboard ? ? colour it As given 62 squares, 31 dominoes. 30 fit; 2 squares are left, apart. Try again, and again. Coloured 30 dark, 32 light. A domino always covers one of each, so 31 can't fit. When you're stuck Stuck signal Same move, three times, nothing new. Done when "A yes with a tiling, or a no with a reason." Change how you see it draw it what never changes try the extreme work backward change the unit split who wants what Drift check Would solving the new version pass "done when"? keep tighten go back Solve, and log the lever No: the colours don't balance. Lever: what never changes. 1 2 3 4 5

No amount of trying tilings proves the answer is no. Colouring the board changes what you're counting, and the answer is two subtractions away. The colours were there all along; the plain drawing hid them.

The system

Try it now.

  1. Set a stuck signal before you start

    Decide now what stuck looks like: the same approach three times with nothing new, or twenty minutes without a new fact. When it fires, stop pushing. More effort inside the same picture finds the same dead ends.

  2. Write the done-when line first

    Before changing anything, write one line from the original problem that says what a solution must do. Every new version gets checked against it, so write it while you still remember what you were asked.

  3. Change how you see it, not how hard you try

    Pull one lever at a time: draw it, find what never changes, try the extreme case, work backward from the end, change the unit, or split it by who wants what.

    Give each lever a fixed stretch, ten minutes or so, and write down anything new it turns up, even if it doesn't solve it.

  4. Check for drift

    Would solving the new version pass the done-when line? Keep it if yes. Tighten it if it covers only part. Go back if you've swapped in an easier problem you like better.

  5. Solve, then log the lever that broke it

    Once it gives, write one line: the problem, the lever, what you saw. Over months that log tells you which levers to try first, and that list is the system.

When to break the rule: if the problem is well defined and you know the method, the answer is more effort, not a new picture. A tax return doesn't need reframing. In an emergency, act on the first sensible reading and restate afterwards. And if a problem belongs to someone else, restate it with them, not for them.

The protocol

Run the stuck protocol on something you're stuck on now.

Write the problem and the done-when line. Count your attempts until the signal fires, pull a lever, then run every new version past the drift check before you work on it. The ledger keeps what you tried.

Stuck signal

Pull one lever

Drift check

    Done when

    Versions tried

      Nothing you type is saved or sent anywhere, and the ledger is gone when you leave the page. Copy the lever that worked into wherever you keep notes.

      Where the levers come from

      On the checkerboard, insight meant finding a better way to see the board, and anything that pointed solvers towards one helped.

      Craig Kaplan and Herbert Simon gave undergraduates the checkerboard on this page and watched them work, timing them, recording them thinking aloud, and building computer simulations of the search. Their argument is that insight is a search for a better representation of the problem, not a better move within the first one. What predicted success was whatever narrowed that search. They tested four kinds.

      • Cue salience

        How noticeable the feature that matters is.

        YouRedraw the problem so the thing you've been ignoring stands out: colour it, count it, put it in a table.

      • Prior knowledge

        What the solver already knows that fits.

        YouAsk where you've met this shape before, in another job, another subject, another year.

      • Hints

        A nudge from outside.

        YouAsk someone for one line pointing somewhere, not for the answer. Give them your done-when line too.

      • Heuristics

        Rules of thumb for where to look.

        YouKeep your own lever list and run it when the stuck signal fires, instead of waiting for inspiration.

      The honest limits: one puzzle family, students in a lab, and a puzzle with a clean answer that real problems rarely have. The study shows what helped people find the representation; it doesn't test a protocol like the one above. The "You" lines are our translation, not the paper's.

      Kaplan & Simon (1990), "In search of insight", Cognitive Psychology 22, 374–419.

      The idea

      Why it works.

      A representation decides which moves exist. On the plain board the only move is placing a domino, and there are too many ways to place them to try. On the coloured board a new move appears, counting, and it settles the question at once. Effort inside the first picture couldn't have found that move, however much there was.

      The risk of changing the picture is that you change the problem with it. That's why the done-when line comes before the levers, and the drift check after: they let you move freely without losing what you were asked.

      How it fails

      Habits that help.

      • Drift. Each version moves a little, and three versions later you're solving something else. Read the done-when line aloud before each new one.
      • Restating into the problem you'd rather have. "How might we market it better" when the product doesn't work. If the new version is mainly more pleasant, the drift check should say go back.
      • Reframing as a way not to start. Four versions and none kept is a pattern. Take the best one and work it for a fixed stretch before you write a fifth.
      • Firing the signal too late. Most people push long past the point where nothing new is coming. Set the signal before you start, while you're still calm about it.

      Go deeper

      Keep going.

      Keep this going in the app.

      HourLife turns these ideas into a daily system — goals, habits, focus, and reflection in one private place.

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