Whiteboard Series and a Puzzle With a Forgotten Constraint
On the whiteboard is the series 3, 7, 15, 31, 63, ___. Devon writes 127 and explains, “Double it and add one: 3 times 2 plus 1 is 7, 7 times 2 plus 1 is 15, and so on.” Priya also writes 127 but argues differently: “The differences are 4, 8, 16 and 32, so the next difference is 64.” Ravi writes 83: “Add 4, then 8, then 12, then 16, then 20.” He checked the first two terms and stopped. Lena writes 64: “Each number is roughly double the one before, so I doubled 63 and rounded down.” At a second station a puzzle card reads: using 2, 3, 4 and 6 exactly once each, with any operations, write an expression equal to 26. Sam writes 6 × 4 + 2 = 26 and moves on. The teacher wants the group to leave with a checking routine they can apply to any series or puzzle, not just the answers to these two.
1. Which series answer is supported? Show exactly where Ravi's rule breaks and why Lena's method cannot count as a rule.
Read the worked answer
127 is supported; Ravi's rule fails at the third step.
Test Ravi term by term:
- 3 + 4 = 7 ✓
- 7 + 8 = 15 ✓
- 15 + 12 = 27 ✗ - the series shows 31, not 27.
His rule matched two steps and then diverged, which is exactly why two matching terms prove nothing. Ravi stopped checking one step too early.
Devon and Priya have produced the same rule in two costumes. “Times 2 plus 1” gives differences that double, and doubling differences force the same result, so their agreement is not a coincidence and both predict 127.
Lena's method is not a rule because it contains no fixed operation. “Roughly double and round down” would give 6 for the first step (3 doubled is 6, not 7), and rounding down leaves the amount of adjustment undecided at every term. A rule must specify one operation that generates every given term exactly; an approximation that happens to land near one term is discarded.
2. Evaluate Sam's puzzle expression against the card's instructions and give a valid expression. What general check does this illustrate?
Read the worked answer
Sam's expression is invalid.
6 × 4 + 2 does equal 26, but it uses only 2, 4 and 6. The card requires all four numbers, each exactly once, so the answer is disqualified on constraint grounds even though the arithmetic is right.
One valid expression is 6 × 3 + 4 × 2 = 18 + 8 = 26, which uses each of 2, 3, 4 and 6 exactly once.
Checking method:
- Does the value match the target? Sam passes this step.
- Does the expression use every listed number exactly once, with no repeats and no extras? Sam fails this step.
Sam optimized for the target and ignored the conditions attached to it, a common pattern in number puzzles. The general habit is to list the constraints before writing anything - target value, which numbers, how many times each, which operations are permitted - then audit the finished expression against the list. An answer that satisfies the most visible condition but breaks a stated one is still wrong.