Problem
Shortest edge routes on a solid
Two equal cubes joined side by side make a wire frame. Every edge is wire, hidden ones included. An ant walks from A to B along the wires by a shortest route. Count the different routes.
Reasoning · Grade 4-2 Counting Shortest Routes
Your answer
Watch out Only the visible faces catch the eye, but the hidden back edges are just as walkable.
How to solve
Strategy Draw a Diagram — First redraw the frame with every hidden edge shown, so that no route can be missed just because a wire is behind the solid. Then notice that the frame is nothing but a flat street grid folded into three directions: from A every useful step is right, back, or down. That makes it the same as the easier flat problem of counting shortest routes on a grid of streets, where you write 1 at the start and then label each corner with the sum of the corners you can arrive from. Finally, a systematic list of the four moves in every order checks the total independently.
1STEP 1
Draw the hidden wires and count the corners
With hidden wires drawn there are 12 corners.
3 × 2 × 2 = 12 corners, 20 wires
Two cubes stuck together share a square face, and drawing every wire — including the ones a solid box would hide — is what stops routes from being overlooked.
1.G.A.2Draw A Diagram2STEP 2
Work out how long a shortest route is
A shortest route uses 4 wires.
2 + 1 + 1 = 4 wires
Adding up how far you must travel in each direction gives the shortest possible number of steps, exactly as it does on a flat map of city blocks.
4.OA.A.3Solve An Easier Related Problem3STEP 3
Use the flat-grid rule: add the routes coming in
As on a grid, add the routes coming in.
(routes to a corner) = (sum of routes to the corners you can come from)
Every route to a corner ends with one last wire, so grouping the routes by which wire they arrive on turns counting into simple addition.
4.OA.C.5Solve An Easier Related ProblemThe number of routes into a corner is the sum of the routes into the corners that feed it.
Why?
Every route into a corner arrives along exactly one of the wires that end there, so the routes split by which wire they used.
🧱Disjoint cases addSplit the possibilities so no case overlaps another and none is left out, and the case counts simply add.
Why?
Each route into a feeding corner extends to exactly one route into this corner, so the counts carry across unchanged.
🧱One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.
4STEP 4
Label the frame corner by corner
Label the corners one by one.
2 + 2 + 2 = 6, 1 + 2 = 3
Filling the labels in order of distance from A guarantees that whenever a corner is reached, all the corners feeding it are already known.
4.OA.A.3Draw A Diagram5STEP 5
Read off the answer at B
The number at B is 12.
6 + 3 + 3 = 12
The last addition collects every route in the frame exactly once, because each route arrives on exactly one of the three final wires.
4.OA.A.3Draw A Diagram6STEP 6
Check by listing the orders of the four moves
Counting the move orders also gives 12.
12 orders of R, R, B, D
Writing the routes as short words and listing them alphabetically makes it obvious that none has been repeated and none forgotten.
4.OA.C.5Make A Systematic ListAnswer
12 routes
6 + 3 + 3 = 12
The answer is a plain count of routes, so it must be a whole number bigger than 1 — and 12 is comfortably in the right range: on a single cube the opposite-corner count is 6, and stretching the box to two cubes should raise it but not explode it, which 12 does. It cannot be more than the 24 ways of ordering four separate moves, and since the two right-steps are interchangeable, exactly half of those orders are duplicates: 24 ÷ 2 = 12. The frame labels also pass a sanity check — no corner's number is smaller than any of the numbers feeding it, and the totals grow steadily 1, 2, 3, 6, 12 as the ant gets further from A.
Takeaway
Draw the hidden wires, write 1 at the start, then keep adding the numbers that flow into each corner — the 3-D frame counts exactly like a flat street map.
- Draw the hidden wires and count the corners
- Work out how long a shortest route is
- Use the flat-grid rule: add the routes coming in
- Label the frame corner by corner
- Read off the answer at B
- Check by listing the orders of the four moves