Cricket Run Rates Analysis

The total score after $N$ overs is $N \times 7.43$, which must be close to an integer. Testing values of $N \in [3, 14]$:

• For $N = 7$, total score $= 7 \times 7.43 = 52.01 \approx 52$. Check consistency:
• $N - 2 = 5$ overs: score $= 5 \times 8.00 = 40$
• $N = 7$ overs: score $= 52$
• $N + 2 = 9$ overs: score $= 9 \times 8.11 = 72.99 \approx 73$
• $N + 4 = 11$ overs: score $= 11 \times 8.45 = 92.95 \approx 93$
• $N + 6 = 13$ overs: score $= 13 \times 8.08 = 105.04 \approx 105$ All over-by-over run totals fall within the valid range of $6$ to $15$ runs per over. Thus $N = 7$.

Scores after overs:

• Over 5: 40
• Over 7: 52 $\implies$ Runs in overs 6 & 7 $= 12$ (must be 6 each since minimum runs per over is 6).
• Over 9: 73 $\implies$ Runs in overs 8 & 9 $= 21$. Since Over 7 $= 6$, max in Over 8 is 15 $\implies$ Runs in overs 7 & 8 max $= 21$.
• Over 11: 93 $\implies$ Runs in overs 10 & 11 $= 20$.
• Runs in overs 9 & 10: Since runs in (8&9) $= 21$ and runs in (10&11) $= 20$, depending on distribution, runs in 9 and 10 can be up to $15 + 10 = 25$, so 22 runs is possible.

Total runs in overs 6 and 7 $= 52 - 40 = 12$. Since minimum runs per over is 6, runs scored in over 6 $= 6$ and runs scored in over 7 $= 6$. 6 is the minimum allowed runs per over, so over 7 must have scored the least possible number of runs.