Think differently: find a reusable pattern
Nine riddles about finding the rule beneath a complicated-looking task. Some have a surprising count; others become simple when you choose a better representation.
Your rhythm: solve one small case → look for a rule → test it on a changed case. Allow 5–15 minutes each. Reveal an answer only after you can explain what you tried.
1. The hidden blocksLink to step
Four positions form a 2 × 2 floor grid. From above, only the front-left and back-left positions contain stacks of unit blocks. Seen straight from the front, the silhouette is one column two blocks high. Must there be exactly three blocks?
Show a hint
Hint: The front and back stacks overlap in the front view. One reaches height two; how high is the other?
Reveal the answer
Answer: No. Both positions contain at least one block. One stack must be two blocks high to produce the silhouette; the other can be one or two blocks high without changing either view. There are three or four blocks. A view from the side would distinguish these cases.
Bring it to your project: Before asking AI to infer from an image, list what its viewpoint cannot show. T09-L01 · Make One Image or Narrated Clip, and Label It is a place to practise that distinction.
Success: You can draw both valid arrangements and name the extra view that would distinguish them.
2. Short chainsLink to step
You have four separate chains, each made of three closed metal links. Opening and reclosing a link costs one operation. How can you make one long chain using only three operations? You may reuse every link.
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Show a hint
Hint: What if one chain supplies connectors instead of staying intact?
Reveal the answer
Answer: Open all three links of one chain. Use those three single links as connectors between the three intact three-link chains. Three opened-and-reclosed links join them into one 12-link chain; you do not need to preserve all four original segments.
Bring it to your project: Can an existing small component serve as connectors among three tools? Explore that before adding a fifth tool in T16-L01 · Pick the Right Tool for One Task by Trying It.
Success: You can count all twelve links in the final chain.
3. Socks in the darkLink to step
A drawer contains many black, white and grey socks, with at least two of each colour. You draw without looking. What is the fewest socks you must draw to guarantee a same-colour pair?
Show a hint
Hint: Construct the least helpful possible first three draws.
Reveal the answer
Answer: Four. The first three could be one of each colour; the fourth must repeat one of those three colours. This is a worst-case guarantee, not a prediction of the most likely draw.
Bring it to your project: A test should still work in the least convenient allowed case. Try that mindset in T03-L03 · Put a Test Set Under the Prompt Your Team Depends On.
Success: You can show both why three may fail and why four cannot fail.
4. Shared birthdaysLink to step
Assume birthdays are independent and uniformly spread over 365 days; ignore leap days. In a room of 23 people, is it more or less likely than not that at least two share a birthday? Estimate before calculating.
Show a hint
Hint: Count the chance that all birthdays differ, then subtract from one. There are many pairs of people.
Reveal the answer
Answer: More likely: about 50.7%. The chance of no match is (365/365) × (364/365) × … × (343/365), about 49.3%, so a match is about 50.7%. The simplified assumptions matter; real birth dates are not exactly uniform.
Bring it to your project: When a small dataset appears surprising, count possible comparisons before calling it an anomaly. See T10-L03 · Build an Analysis Your Colleague Can Rerun and Get the Same Numbers.
Success: You can explain why 23 people create many more than 23 possible matches.
5. Four and seven minutesLink to step
You have a four-minute and a seven-minute hourglass. You can flip either instantly whenever you notice sand finishing. Starting both at the same time, measure exactly nine minutes from the start. You cannot mark the glass or count a minute yourself.
Show a hint
Hint: At minute eight, how much sand is in the seven-minute glass that you flipped at minute seven?
Reveal the answer
Answer: Start both. At minute four, flip the four-minute glass. At minute seven, flip the seven-minute glass. At minute eight, the four-minute glass finishes; flip the seven-minute glass again. Its one minute of sand in the bottom now takes one minute to fall: it finishes at minute nine.
Bring it to your project: Treat observable completion events as triggers, not guesses based on progress bars. T05-L02 · Put AI in the Middle of a Workflow, and Check Every Answer uses that principle.
Success: Your sequence uses only the specified finish events.
6. Make twenty-fourLink to step
Use 8, 8, 3 and 3 exactly once each, with addition, subtraction, multiplication, division and parentheses, to make 24. Fractions are allowed as intermediate results.
Show a hint
Hint: Can you create one third with 3 and 8?
Reveal the answer
Answer: 8 ÷ (3 − 8 ÷ 3) = 24. Inside the parentheses, 3 − 8/3 = 1/3, and 8 ÷ (1/3) = 24. A fractional intermediate value unlocks the result.
Bring it to your project: Name the valid intermediate formats before declaring a solution impossible. In T08-L02 · Get a Script Your Colleague Can Rerun, Without Being a Developer, that means checking what the next step accepts.
Success: You can show each number used once and each permitted operation.
7. One hundred lockersLink to step
One hundred lockers start closed. Person 1 toggles every locker; person 2 toggles every second locker; person 3 every third, and so on through person 100, who toggles only locker 100. Which lockers end open?
Show a hint
Hint: Locker 12 is toggled by one person for each divisor of 12. Which numbers have an odd number of divisors?
Reveal the answer
Answer: The perfect-square lockers: 1, 4, 9, 16, 25, 36, 49, 64, 81 and 100. Most divisors pair up, so those lockers are toggled an even number of times and end closed. A square has one unpaired divisor—its square root—so it is toggled an odd number of times.
Bring it to your project: Instead of simulating a hundred cases, look for the state-changing rule. T08-L02 · Get a Script Your Colleague Can Rerun, Without Being a Developer can then check your prediction.
Success: You can predict whether locker 64 is open without replaying 100 passes.
8. Reach twenty-oneLink to step
Two players start at zero. On each turn, a player adds 1, 2 or 3 to the running total. The first to reach exactly 21 wins; players may not go beyond 21. You play first. Can you guarantee a win?
Show a hint
Hint: Try to leave totals of 1, 5, 9, 13 and 17 after your turns.
Reveal the answer
Answer: Yes. Start by adding 1. Whatever the other player adds next, add enough to make the pair of turns total 4. Your totals after each of your turns are 1, 5, 9, 13, 17, then 21. The rule works because every reply of 1–3 has a complement to 4.
Bring it to your project: Test a strategy against every response, not only the one your assistant chose. See T07-L02 · Hand a Task to an Agent, and Check What It Did.
Success: You can handle all three possible replies after your opening move.
9. Three discsLink to step
Three discs, smallest on top, start on peg A. Move one disc at a time to peg C, using peg B as temporary storage. A larger disc may never rest on a smaller one. What is the minimum number of moves?
Show a hint
Hint: Before you can move the largest disc once, where must the other two be?
Reveal the answer
Answer: Seven moves: move the top two from A to B (3 moves), move the largest from A to C (1), then move the two from B to C (3). For n discs this pattern gives 2ⁿ − 1 moves. The seven-move solution is a small case of the repeatable rule.
Bring it to your project: Break a large task into a smaller copy of the same task, but verify that the smaller task actually finishes. T08-L03 · Write the Spec First, So an AI Assistant Can Change Code Your Team Depends On makes the stopping condition explicit.
Success: You can list the legal seven moves and explain why fewer cannot suffice.
Take one insight back
Choose one rule from this journey and change an input. Does your explanation still work, or did it only fit the first case?
Sources and further practice
Independently worded teaching instances based on hidden block projections (opens in a new tab), short chains (opens in a new tab), matching socks (opens in a new tab), birthday problem (opens in a new tab), hourglasses (opens in a new tab), 24 from four numbers (opens in a new tab), locker puzzle (opens in a new tab), Got It! (opens in a new tab) and Tower of Hanoi (opens in a new tab). The 2 × 2 block layout and target 21 are explicit classroom variants.