Cross every bridge exactly once

Six rooms sit in a three-by-two plan: upper-left, lower-left, upper-middle, lower-middle, upper-right, lower-right. Nine inside doors join them - upper-left to upper-middle, lower-left to upper-middle, lower-left to lower-middle, upper-middle to upper-right, lower-middle to upper-right, lower-middle to lower-right, and one more door joining upper to lower on the left, in the middle and on the right. Every door must be used exactly once. Find which room the walk has to start in.

Your answer

How to solve

Strategy Draw a Diagram — The floor plan is full of things that do not matter — how big the rooms are, where in a wall a door sits, whether a wall is long or short. All that matters is which rooms a door joins. So I redraw the whole building as a dot for each room and a line for each door. Once it is a dot-and-line picture, the question 'can I trace every line exactly once, and where must I start?' is answered just by counting the lines at each dot and checking whether that count is odd or even.

1STEP 1

Name the six rooms

Give the six rooms names.

rooms: UL, LL, UM, LM, UR, LR

2STEP 2

List all nine doors

List all nine doors.

3 + 3 + 3 = 9 doors

3STEP 3

Redraw the building as dots and lines

Redraw rooms as dots and doors as lines.

6 dots, 9 lines

4STEP 4

Count the doors of each room

The door counts are 2, 3, 4, 4, 3, 2.

UL : 2, LL : 3, UM : 4, LM : 4, UR : 3, LR : 2 (2+3+4+4+3+2 = 18 = 2 × 9)

5STEP 5

See why odd rooms have to be the ends

In-and-out pairing makes every middle room even.

in + out pairs → even; only the two ends may be odd

6STEP 6

Find the odd rooms

The odd rooms are lower-left and upper-right.

odd rooms = {LL, UR}

7STEP 7

Walk one actual route

So she starts in the lower-left room.

LL → UM → UL → LL → LM → UR → UM → LM → LR → UR

Answer

the lower-left room

2, 3, 4, 4, 3, 2

The door counts 2, 3, 4, 4, 3, 2 add to 18, which is twice the 9 doors — so no door was missed or double-counted. The number of odd rooms came out as 2, and it always has to be an even number of odd rooms, so that is consistent. The route itself lists 9 steps and therefore visits 10 rooms in sequence, which is right for a walk that crosses 9 doors, and each of the 9 listed doors appears exactly once. Finally, the route starts at one odd room and ends at the other, matching the rule found in step 5.

Takeaway

Turn rooms into dots and doors into lines, then count: a room with an odd number of doors has to be where you start or where you stop.

Name the six rooms
List all nine doors
Redraw the building as dots and lines
Count the doors of each room
See why odd rooms have to be the ends
Find the odd rooms
Walk one actual route

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