Pavel made a list of 31 chores he needed to complete over the next 7 days. On the first 5...

Phase 1: Owning the Dataset

Let's visualize this problem to make it crystal clear...

Timeline of Pavel's Chores:

Day:    1    2    3    4    5    6    7
Chores: 3    5    7    2    6    ?    ?
                                 └─Total: 8─┘

We know:

• Days 1-5: Completed 3, 5, 7, 2, and 6 chores respectively
• Days 6-7: Completed remaining 8 chores total (but split unknown)
• Total verification: \(3 + 5 + 7 + 2 + 6 + 8 = 31\) ✓

Phase 2: Understanding the Question

We need to find the least and greatest possible medians for the 7 daily values.

For the median of 7 numbers:

• We arrange all values in order
• The median is the 4th value (middle of 7)

Current known values in order: 2, 3, 5, 6, 7

Key insight: Days 6 and 7 must add to 8, but we can choose how to split these 8 chores.

Phase 3: Finding the Answer

Possible ways to split 8 chores between days 6 and 7:

1. \(0 + 8 = 8\)
2. \(1 + 7 = 8\)
3. \(2 + 6 = 8\)
4. \(3 + 5 = 8\)
5. \(4 + 4 = 8\)
6. \(6 + 2 = 8\)
7. \(7 + 1 = 8\)
8. \(8 + 0 = 8\)

Let's check key scenarios:

For 4 + 4 split:
All values: 2, 3, 4, 4, 5, 6, 7
Median (4th value) = 4

For 0 + 8 split:
All values: 0, 2, 3, 5, 6, 7, 8
Median (4th value) = 5

For 3 + 5 split:
All values: 2, 3, 3, 5, 5, 6, 7
Median (4th value) = 5

Testing reveals: Only the (4,4) split gives median = 4. All other splits give median = 5.

Phase 4: Solution

Final Answer:

• Least possible median: 4 (when days 6 and 7 each have 4 chores)
• Greatest possible median: 5 (for all other valid splits)