Fire Hydrants
Grid:
×
Hose:
Total Roads
0
Hydrants
0
Roads Covered
0 / 0
Click intersections to place hydrants
Getting Started
1
On the 4×2 grid with 100m hoses, the example uses 8 hydrants. Can you cover all the roads with fewer? What is the minimum?
2
How did you go about finding an arrangement with fewer hydrants? Describe your strategy.
3
Is your solution unique? Can you find a different arrangement using the same minimum number?
Investigating Different Grid Sizes
4
Find the minimum number of hydrants for:
3×2, 4×2, 5×3, 6×4, 5×5. Record your results.5
Can you spot a pattern in your results? How does the minimum number relate to the grid dimensions?
6
Make a prediction for a grid you haven’t tried (e.g. 8×6). Then test it. Were you right?
7
Can you find a formula for the minimum number of hydrants on a
c × r grid with 100m hoses?
Hint: think about how many intersections there are and how many each hydrant can “serve”.
Changing the Hose Length
8
Switch to 200m hoses. How does the minimum number of hydrants change for the same grid? Why?
9
Now try 300m hoses. Each hydrant can now reach 3 blocks along the roads. How few hydrants do you need for a 6×4 grid?
10
For a 5×5 grid, find the minimum hydrants needed for each hose length (100m, 200m, 300m). How does doubling the hose length affect the minimum?
Patterns and Proof
11
For square grids (2×2, 3×3, 4×4, 5×5, 6×6) with 100m hoses, find the minimum each time. Is there a pattern?
12
Can you prove that your answer is truly the minimum? One approach: count the total number of roads and work out the maximum number one hydrant can cover.
13
Is the minimum always unique? For which grid sizes are there multiple optimal arrangements?
Extensions
14
If you have exactly 100 blocks to arrange, which rectangle shape (e.g. 100×1, 50×2, 25×4, 20×5, 10×10) uses the fewest hydrants?
15
What about a city with rectangular blocks (e.g. 200m × 100m instead of 100m × 100m)? How would this change the problem?
16
The cost of installing a hydrant is £5,000 and upgrading hoses from 100m to 200m costs £500 per hydrant. For a 10×10 city, is it cheaper to use more 100m hydrants or fewer 200m ones?
Fire Hydrants
The Problem
A city has streets laid out in a grid. All blocks are 100m squares. The fire department has hoses that are 100m long, so a hydrant at any intersection can reach all roads within 100m. Your job: find the fewest hydrants that cover every road.
How to Use
Click any intersection (dot) to place or remove a hydrant. Green roads are covered; red roads are uncovered. Try to turn all roads green using as few hydrants as possible. Record your best result, then try a different grid size.