Problem
Figures turned through 180 degrees
A rectangle is ruled into 8 squares, 4 across and 2 down. Exactly 4 of them are shaded. A half turn must leave the pattern looking the same. Find every such shading.
Reasoning · Grade 5-2 Line-Symmetric and Point-Symmetric Figures
Your answer
Watch out Picking squares one by one fails: they come with their partners, so you pick pairs.
How to solve
Strategy Make a Systematic List — Guessing shadings one at a time would be slow and I could never be sure I had them all. Instead I first work out what the half turn does to each individual square. That regroups the 8 squares into 4 partner pairs, and the whole question turns into the much smaller question 'which 2 of the 4 pairs do I shade?' A short systematic list of pairs of pairs then gives every answer with nothing missed and nothing repeated.
1STEP 1
Name the squares and find the centre of the turn
Name the squares and find the centre of the turn.
squares = (1,1),(1,2),(1,3),(1,4),(2,1),(2,2),(2,3),(2,4)
Ruling a rectangle into rows and columns of equal squares is a Grade 2 idea, and naming a square by its row and column is exactly how you find a seat in a theatre — no new skill is needed to label the picture.
2.G.A.2Draw A Diagram2STEP 2
See where the half turn sends each square
A half turn sends a top square to the opposite bottom one.
(1,c) ⟶ (2, 5-c)
A half turn is the one rotation you can check with your hands: turn the page upside down and the top-left square is now where the bottom-right square used to be. Nothing changes size or shape, only position.
8.G.A.1Visualize Spatial Relationships3STEP 3
Regroup the 8 squares into 4 partner pairs
The eight squares fall into four pairs.
A={(1,1),(2,4)}, B={(1,2),(2,3)}, C={(1,3),(2,2)}, D={(1,4),(2,1)}
This is the whole trick: stop thinking about 8 separate squares and think about 4 pairs instead. Reorganising the information is what makes an eight-square puzzle into a four-item puzzle.
8.G.A.1Organize Information In More WaysRegrouping the eight squares into four partner pairs turns an eight-square puzzle into a four-item one.
Why?
A half turn carries each square onto exactly one other square, so the squares fall into pairs with none left alone.
🧱Rigid motion preserves length and angleSliding, turning, or flipping a shape lays it exactly onto a copy, so every length and angle stays the same (folding is the flip case).
Why?
Because the pairing is one for one, shading a square forces its partner to be shaded too and the pair behaves as one item.
🧱One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.
4STEP 4
Turn the shading rule into a rule about pairs
Shading four squares means choosing two pairs.
4 ÷ 2 = 2 pairs shaded, out of 4 pairs
Dividing 4 shaded squares into groups of 2 is Grade 3 division, and it converts the shading condition into a simple counting question that a fourth grader can finish in one line.
8.G.A.1Organize Information In More Ways5STEP 5
List every way to choose 2 pairs from the 4
Choosing two of four pairs gives 6 ways.
3 + 2 + 1 = 6
Listing in alphabetical order and counting 3 + 2 + 1 is the standard 'handshake' way of pairing things up — no formula is needed, only the discipline to work through the letters in order.
4.OA.A.3Make A Systematic List6STEP 6
Draw the six shadings
Drawing them shows all six survive the turn.
AB: (1,1),(1,2),(2,3),(2,4) AC: (1,1),(1,3),(2,2),(2,4) AD: (1,1),(1,4),(2,1),(2,4)
Once you know which pairs are shaded, drawing the picture is just colouring in — the thinking has already been done, and the six pictures exactly fill the six blank grids the book provides.
4.G.A.3Draw A DiagramAnswer
6 ways
3 + 2 + 1 = 6
Every one of the six pictures has exactly 4 shaded squares out of 8, so exactly half the sheet is shaded, which matches the instruction. Turning any of the six upside down really does give the same picture back: the shaded squares always come in opposite pairs through the centre. The count is also sensible in size - there are only 4 pairs to choose from, so the answer had to be a small number, and 6 is exactly the number of blank grids the book prints, which is a strong sign that nothing was missed. Finally, 6 is far smaller than the 70 ways of shading any 4 of the 8 squares without the symmetry rule, which is what you would expect once a strong condition is added.
Takeaway
A half turn glues each square to the square opposite it through the centre, so stop counting squares and start counting pairs - then it is just 'pick 2 of the 4 pairs', and there are 6 ways.
- Name the squares and find the centre of the turn
- See where the half turn sends each square
- Regroup the 8 squares into 4 partner pairs
- Turn the shading rule into a rule about pairs
- List every way to choose 2 pairs from the 4
- Draw the six shadings