Wednesday
Wednesday: Puzzle Lab: Two Envelopes
Rocket checks the hook, then hangs the balance at chest height. On the bench sit six envelopes and a jar of beans.
"Puzzle Lab," he says. "Two or three envelopes, all hiding the same number, plus some loose beans. I solve it in two steps."
Raven seals two envelopes, drops them in the left cup with 5 beans, and levels the right cup. "Eleven."
"2x + 5 = 11. What do you notice?" Rocket asks himself, slowly. "The 5 came last, so it goes first."
He lifts 5 beans from each cup, then shares 6 between 2. "Three each." Raven opens an envelope. Three beans.
Nova hovers above the log, her light pulsing. "Write the steps, not just the answer," she says. Nova hums. "Then swap."
Your mission
Play the two-envelope game on the balance. One player hides the same number in each envelope and adds loose beans. The other undoes in two steps.
Log every round: envelopes, loose beans, beans on the right, the equation, both undo steps, your answer and the real count.
- The coat-hanger balance from week 4, on a hook a grown-up has checked.
- About 40 dried beans or buttons, and four to six envelopes.
- String to tie two envelopes into a bundle for a p(x + q) round.
- Your Puzzle Notebook and a pencil.
No one swings on the balance or hangs anything from the hook but the hanger.
Small beans and buttons stay away from children under three and pets. Nothing goes in a mouth.
Shoes on in the garage, and pick up dropped beans so no one slips.
- One player hides the same number of beans in two or three envelopes, plus a few loose beans, in the left cup.
- The other player levels the right cup with beans, counts them, and writes the equation px + q = r.
- Undo the loose beans first on both sides, then share the rest among the envelopes. Write each step.
- Open one envelope and count. Mark the round right or wrong in the log.
- Try a bundle round: tie each envelope to 2 beans, use two bundles, and write p(x + q) = r.
- Swap roles after every round. Log at least six rounds.
The crew's log
| Round | Left cup | Right cup | Equation | After step 1 | Solved |
|---|---|---|---|---|---|
| 1 | 2 envelopes + 5 beans | 11 beans | 2x + 5 = 11 | 2x = 6 | x = 3 |
| 2 | 3 envelopes + 2 beans | 11 beans | 3x + 2 = 11 | 3x = 9 | x = 3 |
| 3 | 2 envelopes + 1 bean | 9 beans | 2x + 1 = 9 | 2x = 8 | x = 4 |
| 4 | 4 envelopes + 4 beans | 12 beans | 4x + 4 = 12 | 4x = 8 | x = 2 |
| 5 | 2 bundles (envelope + 2) | 10 beans | 2(x + 2) = 10 | x + 2 = 5 | x = 3 |
| 6 | 3 envelopes + 4 beans | 11 beans | 3x + 4 = 11 | 3x = 7 | x = 7/3 |
These are the crew's own rounds from Raven's log. Round 6 was a trick: 3 envelopes and 4 beans against 11 left 7 beans to share three ways.
The envelopes could not hold 7/3 beans each, so Raven had written that one on paper as a fraction puzzle.
- Read the question.
- Tap your answer.
Round 1: two envelopes and 5 beans balance 11 beans. How many beans are in each envelope?
Round 2: three envelopes and 2 beans balance 11 beans. How many beans are in each envelope?| Statement | True or false? |
|---|---|
| x = 3 is a solution of 2x + 5 = 11. | ? |
| x = 3 is a solution of 3x + 2 = 11. | ? |
| x = 4 is a solution of 4x + 4 = 12. | ? |
| In round 1, the loose beans come off before the envelopes are shared. | ? |
Six rounds undone in two steps each. Tomorrow the crew rewrites expressions to see what a puzzle really says.