Community Garden Plot Allocation
Five households - Ade, Bello, Chen, Diaz and Eze - are to be assigned plots 1 to 5 in a single row, numbered left to right. The association's rules are: Ade must have plot 3; Eze must have a plot at one end of the row; Bello's plot must be immediately to the left of Diaz's; Chen's plot must be to the right of Bello's; and Chen must not have plot 1. No two households may share a plot. The allocation must be finalised at tomorrow's meeting, and the chair wants to know whether the rules produce one arrangement or several.
1. Determine the full allocation from the rules. Show your deductions and test both possible positions for Eze.
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Ade is fixed at plot 3. Eze must be at 1 or 5, so test both. If Eze is at 1, plots 2, 4 and 5 remain for Bello, Diaz and Chen. Bello must be immediately left of Diaz; the pair (2,3) is impossible because Ade occupies 3, so the pair must be (4,5). That leaves Chen at 2. But Chen must be to the right of Bello, and 2 is left of 4, so this branch fails. Therefore Eze is at 5. Plots 1, 2 and 4 remain. The only consecutive pair available for Bello-Diaz is (1,2), leaving Chen at 4. Check: Chen (4) is right of Bello (1), and Chen is not at plot 1. The unique arrangement is Bello, Diaz, Ade, Chen, Eze in plots 1 to 5.
2. Which single rule could be removed without changing the set of valid arrangements? Show your check.
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The redundant rule is 'Chen must not have plot 1'. In the successful branch Chen is forced to plot 4, so the rule never eliminates anything. To confirm, remove it and re-solve: Eze at 1 still forces Bello-Diaz to (4,5) and Chen to 2, which violates the right-of-Bello rule, so that branch still fails; Eze at 5 still forces Bello-Diaz to (1,2) and Chen to 4. The same unique arrangement results. The other rules are each load-bearing: without Ade at 3, the Bello-Diaz pair could be (2,3); without Eze at an end, two arrangements survive; without Bello immediately left of Diaz, the pair could be separated; without Chen right of Bello, the Eze-at-1 branch would yield a second arrangement.
3. Suppose the 'Eze at an end' rule is replaced by 'Eze is not in plot 3'. Would the allocation still be unique? Explain.
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No. The replacement adds nothing, because Ade already occupies plot 3 and plots are exclusive. Enumerate Bello-Diaz pairs: (1,2) or (4,5), since (2,3) and (3,4) use Ade's plot. If (4,5), the remaining plots 1 and 2 must hold Chen and Eze, but Chen must be right of Bello at 4, which is impossible. If (1,2), plots 4 and 5 hold Chen and Eze in either order, and Chen is right of Bello in both. So the valid arrangements are Bello, Diaz, Ade, Chen, Eze and Bello, Diaz, Ade, Eze, Chen. The allocation is not unique. The chair would need an additional rule - for example, fixing Chen or Eze to a specific plot - before the meeting.