Number Analogies and a Mystery Operation: Which Rule Survives Every Line
Three number-analogy cards and one mystery-operation card are on the table. Card one reads 3 : 9 :: 5 : 15 :: 8 : ___. Omar says 14, explaining that “you add 6 each time, because 3 plus 6 is 9.” Card two reads 4 : 7 :: 10 : 13 :: 20 : ___. Jade says 40 because “the second number is nearly double the first,” while Eli says 23. Card three reads 12 : 3 :: 20 : 5 :: 28 : ___. Rosa says 7, noting that 12 divided by 4 is 3. The mystery-operation card reads 3 ⊙ 4 = 10, 5 ⊙ 2 = 12, 6 ⊙ 3 = 15, and asks for 4 ⊙ 5. Nico proposes “add the two numbers and then add 3,” which works on the first line - 3 plus 4 plus 3 is 10 - but not on the second. The teacher asks the group to decide which proposed rule survives every given line, and to say what a rule must do before it is allowed to predict a missing number.
1. Decide card one and card two. What exactly disqualifies Omar's and Jade's proposals?
Read the worked answer
Card one: 24. Card two: 23.
Card one. Omar's rule is addition by 6. Test it on every line: 3 + 6 = 9 ✓, 5 + 6 = 11 ✗ (the card says 15). It matched the first line and failed the second. Multiplication by 3 fits both lines, 3 × 3 = 9 and 5 × 3 = 15, and extends to 8 × 3 = 24. Omar generalized from one example, which is the same as not checking.
Card two. Eli's rule is addition by 3: 4 + 3 = 7 ✓, 10 + 3 = 13 ✓, so 20 + 3 = 23. Jade's proposal fails twice over. It is not an operation, since “nearly double” does not say how much to add, and its nearest precise version, doubling, already fails the first line: 4 doubled is 8, not 7.
A proposal earns the right to predict only after it reproduces every given pair exactly with no per-line adjusting. Two matched lines are the minimum evidence; one is none.
2. Solve the mystery-operation card and show why Nico's rule collapses.
Read the worked answer
4 ⊙ 5 = 13.
Nico's rule, add both numbers then add 3, passes line one and dies on line two: 5 + 2 + 3 = 10, but the card shows 12. A rule that needs a different constant on each line is not a rule.
The surviving rule is double the first number and add the second:
- 3 ⊙ 4: 2 × 3 + 4 = 10 ✓
- 5 ⊙ 2: 2 × 5 + 2 = 12 ✓
- 6 ⊙ 3: 2 × 6 + 3 = 15 ✓
Applied to the question: 2 × 4 + 5 = 13.
Notice what made the difference. The two numbers play different roles - the first is doubled, the second is added as it stands - so any rule that treats them symmetrically, like plain addition or addition plus a constant, is aimed at the wrong structure. Check the given lines in order, and when a proposal fails, do not patch it with an exception; replace it with a rule whose parts are fixed in advance.
3. Rosa's card-three answer needs a check. State her rule precisely, verify it, and explain what would have gone wrong if she had described the relationship loosely as “the second number is smaller.”
Read the worked answer
Rosa is correct: the answer is 7.
Her rule, stated precisely, is that the second number is the first divided by 4:
- 12 ÷ 4 = 3 ✓
- 20 ÷ 4 = 5 ✓
- 28 ÷ 4 = 7 ✓
Every line is reproduced, so the rule may predict.
A loose description such as “the second number is smaller” fails the prediction test because it does not determine an output. Many smaller numbers would satisfy it, so it cannot choose between 7, 14 and 21, and it offers no way to detect a wrong answer. The precise version can be checked against each line and can be extended with confidence.
The pattern across all four cards is the same: a usable rule is one fixed operation that regenerates every given example exactly and produces one value for the blank. Rules that are approximate, that shift their constant line by line, or that rest on a single example cannot do that, no matter how well they fit the first line of the card.