Problem
Deduction table decides a one-to-one matching
Three people each lost one of a pencil, a notebook and an eraser. No item was lost twice. Five classmates each make one statement. Pick the statements that alone settle who lost what.
Reasoning · Grade 6-1 Deductive Logic (1)
Your answer
Watch out More clues feels stronger, but three of them can still leave two matchings.
How to solve
Strategy Use Matrix Logic — 'Who lost what' with a one-to-one pairing is exactly what a deduction table is for: three people down the side, three items across the top, a circle for yes and a cross for no. I draw five separate copies of the same blank table, one per classmate, and fill in only that classmate's statement. Then the one-to-one rule does the rest of the work automatically — a circle empties its row and its column, and a row or column with two crosses forces a circle in the remaining cell. For the statements that do not finish, I do not merely say 'stuck': I make a systematic list of the matchings that survive and show two different ones, which is the only honest proof that the statement was not enough.
1STEP 1
Set up the table and note there are only six matchings in all
There are 6 possible matchings.
3 × 2 × 1 = 6 possible matchings
Multiplying 3 by 2 by 1 to count the ways of handing out three different items is straightforward Grade 3 group-counting, and knowing there are only six possibilities means every claim in this problem can be checked completely rather than argued about.
3.OA.A.1Make A Systematic List2STEP 2
The two rules that make a table finish itself
One tick in a row clears the rest.
Organising three people against three categories in one small grid is Grade 1 data work; what makes it powerful is that the one-to-one rule turns each new mark into more marks for free.
1.MD.C.4Introduce A VariableTwo rules make the table finish itself: a filled box empties its row and column, and a row with one space left is decided.
Why?
Each person takes exactly one thing and each thing goes to exactly one person, so a match uses up a whole row and a whole column.
🧱One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.
Why?
When crossing out leaves a single open box in a row, that box has to be the match and no guessing is involved.
🧱Rule out the rest and the last one is forcedIf every possibility but one has been ruled out, the survivor must be the answer — no further checking is needed.
3STEP 3
Hannah: not enough — three matchings survive
The first leaves three matchings — not enough.
Two crosses scattered in different rows and different columns can never trigger either rule, so you can tell at a glance that this table will stall before you even list the survivors.
1.MD.C.4Eliminate Possibilities4STEP 4
Daniel: enough — the table fills itself in
The second makes the table fill itself.
Jordan → eraser, Simon → pencil, Mia → notebook
One circle is worth four crosses in a 3 by 3 one-to-one table, which is why a statement that names even one pairing outright usually finishes the job.
1.MD.C.4Introduce A Variable5STEP 5
Chloe: not enough — two matchings survive
The third leaves two matchings.
This is the classic near miss: half of the statement is real information and the other half merely repeats what the first half already forced, so a whole 2 by 2 corner of the table is left blank.
1.MD.C.4Eliminate Possibilities6STEP 6
Ryan: enough — two circles leave only one place for the third
The fourth's two ticks force the third.
Jordan → pencil, Simon → notebook, Mia → eraser
Once two of three people are matched, the last person has no choice left — with a one-to-one pairing you never have to be told the final pair, you just take what is left over.
1.MD.C.4Introduce A Variable7STEP 7
Emma: not enough — two matchings survive, despite three clues
The fifth gives three clues and still falls short.
More clues is not the same as better clues: three crosses arranged diagonally rule out exactly one item per person and leave the two 'cycles' of the remaining pairing, which is why the count of clues is never the thing to check.
1.MD.C.4Eliminate Possibilities8STEP 8
Collect the verdicts
The sufficient ones are the second and the fourth.
Hannah 3, Daniel 1, Chloe 2, Ryan 1, Emma 2
Sorting the five classmates into 'enough' and 'not enough' by a single number — how many matchings survive — turns a wordy comparison into simple counting.
K.MD.B.3Make A Systematic ListAnswer
Daniel and Ryan
3 × 2 × 1 = 6
Every verdict was checked against all 6 possible matchings, not just argued for, and the counts 3, 1, 2, 1, 2 are all between 1 and 6 as they must be — no statement in the problem is self-contradictory, so none leaves 0. The two statements that work do so for a visible reason: each of them names at least one pairing outright with a circle, while the three that fail consist only of crosses or of one circle plus a redundant cross. Notice also that Daniel and Ryan force different matchings from each other — Daniel gives Jordan the eraser, Ryan gives Jordan the pencil — which is fine, because the statements are tested separately and are never required to agree.
Takeaway
To say a clue is not enough, don't say you got stuck — show two different answers that both fit it.
- Set up the table and note there are only six matchings in all
- The two rules that make a table finish itself
- Hannah: not enough — three matchings survive
- Daniel: enough — the table fills itself in
- Chloe: not enough — two matchings survive
- Ryan: enough — two circles leave only one place for the third
- Emma: not enough — two matchings survive, despite three clues
- Collect the verdicts