Depot Route and Crate Net
A driver at a depot must collect a manifest from Stores, a tool from the Workshop and fuel from the Fuel Point before returning to the Gate. On the depot grid, one square is 200 m. The Gate is at (1,1), Stores at (4,1), Workshop at (4,4) and the Fuel Point at (1,4). Vehicles may travel only along grid lines, and each grid-line segment counts as one square. Two restrictions apply: the bridge on the segment from (4,2) to (4,3) is closed, and the underpass on the segment from (1,2) to (1,3) has a clearance of 2.9 m, while the vehicle is 3.1 m high. The driver starts at the Gate facing north. Separately, a packing crate is marked with a cube net: four squares labelled 1, 2, 3 and 4 in a horizontal row; square 5 is attached above square 2; square 6 is attached below square 2. The storeman needs to know which labelled face will be opposite face 2 when the net is folded, and the driver needs the shortest legal route and final travel direction after a specified turn sequence.
1. Find a shortest legal route that visits Stores, Workshop and the Fuel Point and returns to the Gate. Give the distance in squares and metres.
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Shortest legal leg distances:
- Gate to Stores: 3 squares east along y = 1. No restriction.
- Stores to Workshop: direct north is 3 squares, but the segment (4,2)-(4,3) is closed. Detour via x = 5 or x = 3: 5 squares.
- Workshop to Fuel Point: 3 squares west along y = 4. No restriction.
- Fuel Point to Gate: direct south is 3 squares, but the underpass at (1,2)-(1,3) is too low. Detour via x = 2: 5 squares.
One shortest route is Gate -> Stores (3) -> Workshop (5) -> Fuel Point (3) -> Gate (5) = 16 squares. At 200 m per square, that is 16 x 200 = 3,200 m.
A second equally short route is Gate -> Fuel Point (5) -> Workshop (3) -> Stores (5) -> Gate (3) = 16 squares, also 3,200 m. Either is acceptable; the key is that both restrictions force one 2-square detour each.
2. At Stores the driver is facing east. She turns 90 degrees left, then 180 degrees, then 90 degrees right. What direction is she facing?
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Start facing east.
- 90 degrees left from east = north.
- 180 degrees from north = south.
- 90 degrees right from south = west.
Final direction: west. A useful check is to combine the turns: left 90 + 180 + right 90 is equivalent to 180 degrees net change from east, which gives west.
3. For the cube net, which labelled face is opposite face 2 when folded? Explain the folding.
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Face 4 is opposite face 2.
The four squares 1-2-3-4 form a band around the cube. In such a row, alternate faces are opposite: 1 is opposite 3, and 2 is opposite 4. Squares 5 and 6 are attached above and below square 2, so they become the top and bottom faces and are opposite each other. Therefore the face opposite face 2 is face 4.