Think differently: test the constraints
Eight puzzles for checking a layout, a claim or a plan against its exact rules. Sketch the problem first. A good explanation checks the losing cases too.
Your rhythm: list the constraints → draw or tabulate a small case → test → explain. Allow about 10 minutes each; the two-ball challenge can take longer.
1. Fold the netLink to step
Six equal paper squares join edge to edge in each layout below. Which layout can fold into a closed cube without overlapping faces? Imagine folding along the joined edges; no cutting or rearranging.
Show a hint
Hint: Can all six squares land on different cube faces?
Reveal the answer
Answer: A can. Its vertical strip wraps around four side faces; the two attached squares become top and bottom. B cannot: the 2 × 3 rectangle makes faces collide when folded into a cube. You can check with six paper squares before trusting a mental rotation.
Bring it to your project: Compare how a design looks in a flat plan with how it works once assembled. T06-L02 · Build a Small Shared Tool From a Prompt, and Test It With a Colleague makes that distinction concrete.
Success: You can point to six distinct faces in A and a collision in B.
2. Ten dots, five linesLink to step
Place ten dots so that five straight lines each pass through exactly four dots. A dot may be counted on more than one line. Sketch before opening the answer.
Show a hint
Hint: Let the lines cross. How can one dot serve two lines?
Reveal the answer
Answer: Draw a regular five-pointed star (a pentagram). Put one dot at each of its five outer tips and each of its five inner crossings. Every long stroke of the star passes through two outer tips and two inner crossings: five straight lines, four dots per line.
Bring it to your project: A shared component can satisfy two requirements without duplicating the work. Identify one reusable piece in T17-L03 · Explore What Already Exists Before You Build.
Success: You can count all five lines and all ten distinct dots without adding a hidden sixth line.
3. Count every squareLink to step
Draw a 3 × 3 grid of equal small squares. How many axis-aligned squares of all sizes can you find? A square must follow the drawn grid lines.
Show a hint
Hint: Count the 1 × 1, 2 × 2 and 3 × 3 squares separately.
Reveal the answer
Answer: Fourteen: nine small, four medium and one covering the whole grid. Counting only visible little cells misses five valid squares.
Bring it to your project: State your counting unit before evaluating a result. T10-L03 · Build an Analysis Your Colleague Can Rerun and Get the Same Numbers relies on definitions like this.
Success: You can enumerate each size without counting any square twice.
4. Cross the bridge at nightLink to step
Four people need to cross a narrow bridge with one torch. At most two cross together, and any crossing requires the torch. Their individual crossing times are 1, 2, 5 and 10 minutes; a pair moves at the slower person's pace. What is the least total time?
Show a hint
Hint: The torch must come back twice. Which returns are cheap, and can the two slowest cross together?
Reveal the answer
Answer: Seventeen minutes. The 1- and 2-minute walkers cross (2); 1 returns (1); 5 and 10 cross together (10); 2 returns (2); 1 and 2 cross (2). Total 2 + 1 + 10 + 2 + 2 = 17. Alternatives that send each slow walker separately cost at least 19 under these rules.
Bring it to your project: Include recovery and handoff steps when comparing workflow costs. A local saving may make the whole process slower in T05-L02 · Put AI in the Middle of a Workflow, and Check Every Answer.
Success: Your schedule accounts for every torch return, not just the forward trips.
5. Two speakersLink to step
On an island, each person always tells the truth or always lies. Ada says, “Ben is a liar.” Ben says, “Ada and I are different kinds.” Which kind is each?
Show a hint
Hint: Assume Ada tells the truth and see whether Ben's sentence can then be false.
Reveal the answer
Answer: Ada lies and Ben tells the truth. If Ada spoke truthfully, Ben would lie—but his statement that they differ would then be true, a contradiction. If Ada lies, Ben is truthful and they really differ.
Bring it to your project: Ask an assistant to check its own proposed assignments against every condition, not just one convincing quote. See T03-L03 · Put a Test Set Under the Prompt Your Team Depends On.
Success: You can rule out the alternative assignment explicitly.
6. Two test ballsLink to step
There is a highest floor from which a test ball can be dropped without breaking. All higher floors break it; all lower floors do not. You have two identical balls and a 100-floor building. A ball survives every safe drop and can be reused. How many drops do you need to guarantee finding the threshold in the worst case? Treat floor 0 as safe.
Show a hint
Hint: After a break, only one ball remains, so you must test untried floors one by one. Make each successive jump smaller.
Reveal the answer
Answer: Fourteen drops suffice and 13 cannot guarantee it. Drop first at floor 14, then 27 (13 higher), 39 (12 higher), and so on, reducing the jump by one each time. If the first ball breaks, use the second to check the floors since the last safe drop in order. The worst case stays at 14 tests. Thirteen such steps cover only 13 + 12 + … + 1 = 91 floors; fourteen cover 105, enough for 100.
Bring it to your project: When testing a tool, count the cost of a failed test too. T16-L03 · Test a Tool on Your Own Work Before Your Team Depends on It makes the test budget explicit.
Success: You can explain why the jump sizes decrease and why 13 tests are insufficient.
7. The white marbleLink to step
Bag A contains two white marbles. Bag B contains one white and one black marble. Choose a bag at random (equal chance), then draw one marble at random from it. You see a white marble. What is the chance the other marble in that same bag is white?
Show a hint
Hint: List the equally likely bag-and-marble draws that show white.
Reveal the answer
Answer: Two-thirds. Of the four equally likely draws (A's first white, A's second white, B's white, B's black), three show white. In two of those three cases the bag is A, whose other marble is white. Do not treat the two bag names as equally likely after seeing white.
Bring it to your project: State how a sample was selected before interpreting a percentage. See T10-L01 · Ask a Spreadsheet a Question and Check Which Rows It Used.
Success: You can name the three surviving observations and their counts.
8. Four litresLink to step
You have an unmarked 3-litre jug, an unmarked 5-litre jug, an unlimited water supply and a drain. You may fill either jug, empty either jug, or pour from one into the other until the source is empty or the target is full. How can you leave exactly 4 litres in the 5-litre jug?
Show a hint
Hint: First create a 2-litre remainder in the larger jug.
Reveal the answer
Answer: Fill the 5-litre jug and pour into the 3-litre jug, leaving 2 litres in the 5-litre jug. Empty the 3-litre jug, pour the 2 litres into it, then refill the 5-litre jug. Pour 1 litre from the 5-litre jug to fill the 3-litre jug. The 5-litre jug now holds exactly 4 litres.
Bring it to your project: Write down allowed state changes before automating them. T05-L01 · Your First Automation: Build It, Break It, Switch It Off starts with a small, inspectable flow.
Success: You can list the state of both jugs after each move.
Take one insight back
Pick a claim from your current project. Write one case that would disprove it and one measurement that would be needed to check that case.
Sources and further practice
Original classroom instances adapted from cube nets (opens in a new tab), ten balls in five lines (opens in a new tab), counting squares (opens in a new tab), bridge crossing (opens in a new tab), knights and knaves (opens in a new tab), two test balls (opens in a new tab), bags of marbles (opens in a new tab) and measuring four litres (opens in a new tab). The two-speaker instance uses our own statements; the 3 × 3 grid specifies its own size.