Problem
Rolling a die along a path
A die is tipped one square at a time to the shaded square. Eight little journeys are drawn, two in each of four groups. Opposite faces of the die add to 7. Say how many pips show on top at the end of each.
Reasoning · Grade 5-2 Dice
Your answer
Watch out Counting rolls is not enough — which way it rolled matters too.
How to solve
Strategy Create a Physical Representation — Trying to picture eight tumbling dice in my head is where mistakes happen, so I put a real die on the table and roll it square by square along each drawn arrow. To keep a record I label the six positions — top, bottom, front, back, left, right — and write down which pip number sits in each position after every quarter turn. Doing that once for each shape of route shows a pattern: the shape of the path, not its length, decides the answer. Then I only have to recognise which of four shapes each drawing is (straight line, L, U, or N) and read off the rule, and the long routes become no harder than the short ones.
1STEP 1
Fill in the three hidden faces first
Fill the three hidden faces by subtracting from 7.
7 - 1 = 6, 7 - 2 = 5, 7 - 3 = 4
One subtraction inside 10 turns a picture with three hidden faces into a die you know completely, and knowing the bottom face is what the answers keep depending on.
2.OA.A.1Visualize Spatial Relationships2STEP 2
Learn what one roll actually does
One roll brings a side face to the top.
roll right: left → top → right → bottom → left
A roll is just a quarter turn, and a quarter turn cannot bend or stretch the die, so the six numbers only trade places — that is why a child with one real die can settle any of these questions.
8.G.A.1Create A Physical Representation3STEP 3
Group (1): a straight line, so count the rolls in fours
A straight run repeats every four rolls.
1 → 4 → 6 → 3 → 1
Four quarter turns make one whole turn, so rolling straight repeats with period 4 — after that you only need the remainder when the number of rolls is divided by 4, and 2 rolls always brings the starting bottom face up.
4.OA.C.5Look For A PatternRolling straight along a line brings the same face back on top every four rolls, so the rolls can be counted in fours.
Why?
Four faces take turns on top as the die rolls one way, and after the fourth roll the ring has come all the way round.
🧱A cycle returns after a full periodA fixed repeating cycle of length p comes back to the same position every p steps.
Why?
Only what is left after whole groups of four decides the final top face, so full laps can simply be dropped.
🧱Division with remainderAny number splits into a whole count of equal groups plus a leftover, and the leftover is always smaller than the group size.
4STEP 4
Group (2): an L-shaped route brings a side face to the top
An L-shaped route brings a side face up.
left: top = 3, right: top = 4, 3 + 4 = 7
The first roll never disturbs the two side faces, so after the turn one of them is waiting to be lifted onto the top — you can watch it happen on a real die in two seconds.
8.G.A.1Create A Physical Representation5STEP 5
Group (3): the U-shaped route undoes itself
A U-shaped route undoes itself.
top → front → front → top
Testing the shortest U first — forward then straight back, which obviously changes nothing — shows why the sideways rolls in between are harmless, and then the long route is no harder than the short one.
8.G.A.1Solve An Easier Related Problem6STEP 6
Group (4): the N-shaped route flips the die over
A zigzag route flips the die over.
forward, right, forward: 3 → 1 → 2 → 4 and right, forward, right: 3 → 2 → 1 → 4
Three quarter turns about edges that keep changing direction add up to a half turn of the whole die, and a half turn is exactly what swaps the top face with the bottom face — so the answer is 7 minus the starting top.
8.G.A.1Create A Physical Representation7STEP 7
Collect the eight answers
The eight answers are 6, 1, 3, 4, 5, 5, 4, 4.
(1) 6, 1 (2) 3, 4 (3) 5, 5 (4) 4, 4
Once each route is sorted into one of four shapes, the answers come from a rule rather than from re-rolling the die eight separate times.
4.OA.C.5Look For A PatternAnswer
6 and 1, 3 and 4, 5 and 5, 4 and 4
Every answer is a whole number from 1 to 6, which is all a die can ever show. The two answers inside group (2) are 3 and 4, which add to 7 — exactly right, because the two routes are mirror images of each other, so they must lift opposite faces onto the top. Group (1)'s two answers, 6 and 1, also add to 7, for the same reason: 2 rolls turns the die half way over and 4 rolls turns it all the way round. In groups (3) and (4) the long route and the short route agree with each other, which is what should happen because the extra squares are all straight-ahead sideways rolls that leave the deciding faces untouched. Finally, none of the answers is the face the die was standing on in a way that contradicts the pictures: group (3) keeps its starting 5 and group (4) shows its starting bottom 4.
Takeaway
Roll a real die along the arrow and watch the top face: straight lines repeat every 4 rolls, a U-turn brings the same face back, and a zigzag flips the die right over!
- Fill in the three hidden faces first
- Learn what one roll actually does
- Group (1): a straight line, so count the rolls in fours
- Group (2): an L-shaped route brings a side face to the top
- Group (3): the U-shaped route undoes itself
- Group (4): the N-shaped route flips the die over
- Collect the eight answers