Think Again!18 puzzles. Pick one that catches your eye. Stuck? Try a different move.
Think it through. Try it out.
Almost there, just right.
You have two unmarked jugs: 7 L and 4 L.
Can you leave exactly 5 liters of water in one jug?
Keep pouring until one jug is empty or the other is full.
Fill either jug and see what happens.
That little moment of discovery
5 liters. You made “almost” exactly right.
Even without marks, the difference between jugs can measure for you.
The useful quantity is how much room remains. Use the empty space in one jug to measure water in the other, and you can leave an amount that was never marked.
Pour a full 7-liter jug into the 4-liter jug and 3 liters remain. Empty the small jug, then move those 3 liters into it. It now has room for exactly 1 more liter. That 1-liter gap becomes another measuring tool.
What if the jugs held 8 and 4 liters? Could filling, emptying, and pouring still produce 5 liters?
No. Starting with empty jugs, all three operations keep each amount a multiple of 4, so 5 is unreachable. This is a powerful approach: look not just for the next move, but for a property every move preserves.
My attempts 0 moves
- No moves yet. You can undo every step.
No timer. No need to get it right first time. Go ahead and try.
Every press starts a chain reaction
One press, three lights.
Six lights form a ring.
Try to turn them all off.
Tap a light to switch it and its two neighbors: on becomes off, and off becomes on. Lights 1 and 6 are neighbors too.
When the lights went out, a pattern lit up.
Pressing a light twice cancels out. The order of presses does not matter either. You can summarize a whole sequence by which lights were pressed an odd number of times.
You can always undo. Make the chain reaction your tool.
Think a little more. Weigh a little less.
Two weighings. Find the coin.
8 identical-looking coins. Exactly 1 is heavier.
Use the balance at most twice to identify it for sure.
Choose Left pan or Right pan, then tap coins to move them. Weigh only with equal, nonzero counts on both pans. Once certain, select the heavy coin below and check your judgment.
You found a conclusion backed by evidence.
A balance has three possible outcomes: left heavier, right heavier, or balanced. First split 8 coins into groups of 3, 3, and 2. Then, among at most 3 candidates, weigh 1 against 1. Every result narrows the possibilities.
Each round has exactly one heavier coin. Evidence is more interesting than a lucky guess.
It always seems to work. But does it?
Challenge a pattern.
Someone claims that for every nonnegative integer n,
this expression is always prime. Find an n that produces a number that is not prime.
A prime is greater than 1 and has only two positive factors: 1 and itself.
Enter an integer from 0 to 10,000. Trial division gives exact results throughout this range.
Start at 0 or jump around. Do not trust the pattern too quickly.
My test record 0 tests
One counterexample stops “always.”
No number of successful examples proves a claim about every integer. But a single example where it fails disproves “always.”
Go a little deeper: n² + n = n(n + 1). When is this part also divisible by 11?
Go beyond guessing. Test a claim for yourself.
Pick a puzzle. Clear it to open an extra step and an extreme challenge outside the ring. Wander off to another puzzle whenever you like.
