Math Detours · Grade 5 · Around the corner · 15

Who owns the fish?

Four children, four pets, a handful of clues. A grid of ticks and crosses turns guessing into certainty.

Name ____________________Date ____________

Tonight's idea

Eli, Fay and Gus each play one instrument: the drum, the flute or the guitar. No two play the same one. There are two clues. (1) Eli does not play the drum. (2) Gus does not play the drum or the flute.

Draw a grid. Each row is a person and each column is an instrument. A cross means "no". A tick means "yes".

drum flute guitar Eli Fay Gus drum flute guitar Eli Fay Gus after the cluesfinished
Blue crosses come from the clues. The grey crosses and the ticks come from the rules.

Two rules do the work. Every row gets exactly one tick, because each person plays one instrument. Every column gets exactly one tick, because each instrument has one player.

When a row or column has one empty box left, that box is a tick. After a tick, cross out the rest of its row and its column.

You never guess. A clue or a rule forces every mark.

Watch one: Eli, Fay and Gus

The clues give three crosses. Gus's row has one empty box, so Gus plays the guitar. Tick it, and cross out the rest of the guitar column.

Now Eli's row has one empty box. Eli plays the flute. Cross out the rest of the flute column. Fay is left with the drum.

Last, read the clues again and check the answer against each one. Both hold.

Your turn

  1. A grid has four children and four pets: cat, dog, hamster and parrot. Mia's row has crosses under cat, dog and hamster. (a) What goes in Mia's parrot box? (b) What goes in the other three boxes of the parrot column?
  2. Hal, Ivy and Jo each ride one thing: a bike, a scooter or a skateboard. Hal does not ride the scooter. Ivy does not ride the scooter or the skateboard. Draw a grid and find who rides what.
  3. Who owns the fish? Ana, Ben, Cal and Dee each own one pet: a cat, a dog, a fish or a rabbit. (1) Ana does not own the cat or the fish. (2) Ben does not own the cat, the fish or the rabbit. (3) Dee does not own the fish. Fill in the grid.
    cat dog fish rabbit Ana Ben Cal Dee
  4. Rub out clue 3 in question 3. Find two different answers that fit clues 1 and 2. Why does that make it a bad puzzle?

Stretch: no clues at all

With no clues, how many ways can four children own four different pets? Let Ana choose first, then Ben, then Cal, then Dee. How many of those ways fit clue 2 from question 3?

Answers

  1. (a) A tick. It is the only box left in Mia's row, so Mia owns the parrot. (b) Three crosses. Nobody else can own the parrot.
  2. Ivy's row has one empty box: Ivy rides the bike. Cross out the rest of the bike column. Hal cannot ride the bike or the scooter, so Hal rides the skateboard. Jo rides the scooter.
  3. Clue 2 leaves one empty box in Ben's row: Ben owns the dog. Cross out the rest of the dog column. Ana cannot own the cat, the dog or the fish, so Ana owns the rabbit. Dee cannot own the dog, the fish or the rabbit, so Dee owns the cat. Cal owns the fish.
  4. Ben still owns the dog and Ana still owns the rabbit. The cat and the fish are left for Cal and Dee, either way round: Cal fish and Dee cat, or Cal cat and Dee fish. The clues cannot tell you which is right, so you would have to guess.

Stretch. Ana has 4 choices, Ben has 3, Cal has 2 and Dee has 1. \(4 \times 3 \times 2 \times 1 = 24\) ways. Clue 2 gives Ben the dog. The other three children share three pets in \(3 \times 2 \times 1 = 6\) ways.

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