Problem Analysis #39

16.47s
14,865 toks
Problem Statement

Ten empty boxes are arranged in a row, and there are eight balls. The balls are to be placed into the boxes so that no boxes are left unused, the total number of balls is 88, and each box contains at most two balls. (The balls are indistinguishable.) Find the number of such distributions that satisfy the following conditions. [4 points]\begin{center}\fbox{\begin{minipage}{0.9\textwidth}(A) The number of boxes containing exactly one ball is either 44 or 66.(B) Any box containing two balls has empty neighboring boxes.\end{minipage}}\end{center}\

Ground Truth
262
Execution Trace
Reasoning Process

[THOUGHT] Let me analyze this problem step by step.

We have 10 boxes in a row and 8 balls to distribute. Each box can contain 0, 1, or 2 balls. We need:

  1. No boxes left unused means every box has at least one ball? Wait, that doesn't make sense with 10 boxes and 8 balls.

Let me re-read: "no boxes are left unused" - this might mean something different. Actually, looking at the conditions more carefully:

Condition (A): The number of boxes containing exactly one ball is either 4 or 6. Condition (B): Any box containing two balls has empty neighboring boxes.

Wait, if there are empty neighboring boxes, then some boxes ARE empty. So "no boxes are left unused" must mean something else.

Let me reconsider: Perhaps "no boxes are left unused" means we must use all 10 boxes in our arrangement (i.e., we can't just use fewer boxes). Or it could be a translation issue.

Given the constraints:

  • Total balls = 8
  • Let's say there are x boxes with 2 balls, y boxes with 1 ball, and z boxes with 0 balls
  • x + y + z = 10
  • 2x + y = 8
  • Condition A: y = 4 or y = 6

If y = 4: 2x + 4 = 8, so x = 2, and z = 10 - 2 - 4 = 4 If y = 6: 2x + 6 = 8, so x = 1, and z = 10 - 1 - 6 = 3

Condition B: Any box with 2 balls must have empty neighbors on both sides.

Let me write code to enumerate all valid configurations. [/THOUGHT] [PYTHON] from itertools import product

def count_valid_distributions(): count = 0 valid_configs = []

# Each box can have 0, 1, or 2 balls
# We need to check all configurations of 10 boxes
for config in product([0, 1, 2], repeat=10):
    # Check total balls = 8
    if sum(config) != 8:
        continue
    
    # Count boxes with exactly 1 ball
    ones = config.count(1)
    
    # Condition A: ones must be 4 or 6
    if ones not in [4, 6]:
        continue
    
    # Condition B: Any box with 2 balls must have empty neighbors
    valid = True
    for i in range(10):
        if config[i] == 2:
            # Check left neighbor
            if i > 0 and config[i-1] != 0:
                valid = False
                break
            # Check right neighbor
            if i < 9 and config[i+1] != 0:
                valid = False
                break
    
    if valid:
        count += 1
        valid_configs.append(config)

return count, valid_configs

count, configs = count_valid_distributions() print(f"Total count: {count}") print("\nValid configurations:") for c in configs: ones = c.count(1) twos = c.count(2) print(f"{c} - ones: {ones}, twos: {twos}") [/PYTHON]

Tool Output
[PYTHON OUTPUT] Total count: 262 Valid configurations: (0, 0, 1, 1, 1, 1, 0, 2, 0, 2) - ones: 4, twos: 2 (0, 0, 1, 1, 1, 1, 1, 1, 0, 2) - ones: 6, twos: 1 (0, 0, 2, 0, 1, 1, 1, 1, 0, 2) - ones: 4, twos: 2 (0, 0, 2, 0, 1, 1, 1, 1, 1, 1) - ones: 6, twos: 1 (0, 0, 2, 0, 2, 0, 1, 1, 1, 1) - ones: 4, twos: 2 (0, 1, 0, 1, 1, 1, 0, 2, 0, 2) - ones: 4, twos: 2 (0, 1, 0, 1, 1, 1, 1, 1, 0, 2) - ones: 6, twos: 1 (0, 1, 0, 2, 0, 1, 1, 1, 0, 2) - ones: 4, twos: 2 (0, 1, 0, 2, 0, 1, 1, 1, 1, 1) - ones: 6, twos: 1 (0, 1, 0, 2, 0, 2, 0, 1, 1, 1) - ones: 4, twos: 2 (0, 1, 1, 0, 1, 1, 0, 2, 0, 2) - ones: 4, twos: 2 (0, 1, 1, 0, 1, 1, 1, 1, 0, 2) - ones: 6, twos: 1 (0, 1, 1, 0, 2, 0, 1, 1, 0, 2) - ones: 4, twos: 2 (0, 1, 1, 0, 2, 0, 1, 1, 1, 1) - ones: 6, twos: 1 (0, 1, 1, 0, 2, 0, 2, 0, 1, 1) - ones: 4, twos: 2 (0, 1, 1, 1, 0, 1, 0, 2, 0, 2) - ones: 4, twos: 2 (0, 1, 1, 1, 0, 1, 1, 1, 0, 2) - ones: 6, twos: 1 (0, 1, 1, 1, 0, 2, 0, 1, 0, 2) - ones: 4, twos: 2 (0, 1, 1, 1, 0, 2, 0, 1, 1, 1) - ones: 6, twos: 1 (0, 1, 1, 1, 0, 2, 0, 2, 0, 1) - ones: 4, twos: 2 (0, 1, 1, 1, 1, 0, 0, 2, 0, 2) - ones: 4, twos: 2 (0, 1, 1, 1, 1, 0, 1, 1, 0, 2) - ones: 6, twos: 1 (0, 1, 1, 1, 1, 0, 2, 0, 0, 2) - ones: 4, twos: 2 (0, 1, 1, 1, 1, 0, 2, 0, 1, 1) - ones: 6, twos: 1 (0, 1, 1, 1, 1, 0, 2, 0, 2, 0) - ones: 4, twos: 2 (0, 1, 1, 1, 1, 1, 0, 1, 0, 2) - ones: 6, twos: 1 (0, 1, 1, 1, 1, 1, 0, 2, 0, 1) - ones: 6, twos: 1 (0, 1, 1, 1, 1, 1, 1, 0, 0, 2) - ones: 6, twos: 1 (0, 1, 1, 1, 1, 1, 1, 0, 2, 0) - ones: 6, twos: 1 (0, 2, 0, 0, 1, 1, 1, 1, 0, 2) - ones: 4, twos: 2 (0, 2, 0, 0, 1, 1, 1, 1, 1, 1) - ones: 6, twos: 1 (0, 2, 0, 0, 2, 0, 1, 1, 1, 1) - ones: 4, twos: 2 (0, 2, 0, 1, 0, 1, 1, 1, 0, 2) - ones: 4, twos: 2 (0, 2, 0, 1, 0, 1, 1, 1, 1, 1) - ones: 6, twos: 1 (0, 2, 0, 1, 0, 2, 0, 1, 1, 1) - ones: 4, twos: 2 (0, 2, 0, 1, 1, 0, 1, 1, 0, 2) - ones: 4, twos: 2 (0, 2, 0, 1, 1, 0, 1, 1, 1, 1) - ones: 6, twos: 1 (0, 2, 0, 1, 1, 0, 2, 0, 1, 1) - ones: 4, twos: 2 (0, 2, 0, 1, 1, 1, 0, 1, 0, 2) - ones: 4, twos: 2 (0, 2, 0, 1, 1, 1, 0, 1, 1, 1) - ones: 6, twos: 1 (0, 2, 0, 1, 1, 1, 0, 2, 0, 1) - ones: 4, twos: 2 (0, 2, 0, 1, 1, 1, 1, 0, 0, 2) - ones: 4, twos: 2 (0, 2, 0, 1, 1, 1, 1, 0, 1, 1) - ones: 6, twos: 1 (0, 2, 0, 1, 1, 1, 1, 0, 2, 0) - ones: 4, twos: 2 (0, 2, 0, 1, 1, 1, 1, 1, 0, 1) - ones: 6, twos: 1 (0, 2, 0, 1, 1, 1, 1, 1, 1, 0) - ones: 6, twos: 1 (0, 2, 0, 2, 0, 0, 1, 1, 1, 1) - ones: 4, twos: 2 (0, 2, 0, 2, 0, 1, 0, 1, 1, 1) - ones: 4, twos: 2 (0, 2, 0, 2, 0, 1, 1, 0, 1, 1) - ones: 4, twos: 2 (0, 2, 0, 2, 0, 1, 1, 1, 0, 1) - ones: 4, twos: 2 (0, 2, 0, 2, 0, 1, 1, 1, 1, 0) - ones: 4, twos: 2 (1, 0, 0, 1, 1, 1, 0, 2, 0, 2) - ones: 4, twos: 2 (1, 0, 0, 1, 1, 1, 1, 1, 0, 2) - ones: 6, twos: 1 (1, 0, 0, 2, 0, 1, 1, 1, 0, 2) - ones: 4, twos: 2 (1, 0, 0, 2, 0, 1, 1, 1, 1, 1) - ones: 6, twos: 1 (1, 0, 0, 2, 0, 2, 0, 1, 1, 1) - ones: 4, twos: 2 (1, 0, 1, 0, 1, 1, 0, 2, 0, 2) - ones: 4, twos: 2 (1, 0, 1, 0, 1, 1, 1, 1, 0, 2) - ones: 6, twos: 1 (1, 0, 1, 0, 2, 0, 1, 1, 0, 2) - ones: 4, twos: 2 (1, 0, 1, 0, 2, 0, 1, 1, 1, 1) - ones: 6, twos: 1 (1, 0, 1, 0, 2, 0, 2, 0, 1, 1) - ones: 4, twos: 2 (1, 0, 1, 1, 0, 1, 0, 2, 0, 2) - ones: 4, twos: 2 (1, 0, 1, 1, 0, 1, 1, 1, 0, 2) - ones: 6, twos: 1 (1, 0, 1, 1, 0, 2, 0, 1, 0, 2) - ones: 4, twos: 2 (1, 0, 1, 1, 0, 2, 0, 1, 1, 1) - ones: 6, twos: 1 (1, 0, 1, 1, 0, 2, 0, 2, 0, 1) - ones: 4, twos: 2 (1, 0, 1, 1, 1, 0, 0, 2, 0, 2) - ones: 4, twos: 2 (1, 0, 1, 1, 1, 0, 1, 1, 0, 2) - ones: 6, twos: 1 (1, 0, 1, 1, 1, 0, 2, 0, 0, 2) - ones: 4, twos: 2 (1, 0, 1, 1, 1, 0, 2, 0, 1, 1) - ones: 6, twos: 1 (1, 0, 1, 1, 1, 0, 2, 0, 2, 0) - ones: 4, twos: 2 (1, 0, 1, 1, 1, 1, 0, 1, 0, 2) - ones: 6, twos: 1 (1, 0, 1, 1, 1, 1, 0, 2, 0, 1) - ones: 6, twos: 1 (1, 0, 1, 1, 1, 1, 1, 0, 0, 2) - ones: 6, twos: 1 (1, 0, 1, 1, 1, 1, 1, 0, 2, 0) - ones: 6, twos: 1 (1, 0, 2, 0, 0, 1, 1, 1, 0, 2) - ones: 4, twos: 2 (1, 0, 2, 0, 0, 1, 1, 1, 1, 1) - ones: 6, twos: 1 (1, 0, 2, 0, 0, 2, 0, 1, 1, 1) - ones: 4, twos: 2 (1, 0, 2, 0, 1, 0, 1, 1, 0, 2) - ones: 4, twos: 2 (1, 0, 2, 0, 1, 0, 1, 1, 1, 1) - ones: 6, twos: 1 (1, 0, 2, 0, 1, 0, 2, 0, 1, 1) - ones: 4, twos: 2 (1, 0, 2, 0, 1, 1, 0, 1, 0, 2) - ones: 4, twos: 2 (1, 0, 2, 0, 1, 1, 0, 1, 1, 1) - ones: 6, twos: 1 (1, 0, 2, 0, 1, 1, 0, 2, 0, 1) - ones: 4, twos: 2 (1, 0, 2, 0, 1, 1, 1, 0, 0, 2) - ones: 4, twos: 2 (1, 0, 2, 0, 1, 1, 1, 0, 1, 1) - ones: 6, twos: 1 (1, 0, 2, 0, 1, 1, 1, 0, 2, 0) - ones: 4, twos: 2 (1, 0, 2, 0, 1, 1, 1, 1, 0, 1) - ones: 6, twos: 1 (1, 0, 2, 0, 1, 1, 1, 1, 1, 0) - ones: 6, twos: 1 (1, 0, 2, 0, 2, 0, 0, 1, 1, 1) - ones: 4, twos: 2 (1, 0, 2, 0, 2, 0, 1, 0, 1, 1) - ones: 4, twos: 2 (1, 0, 2, 0, 2, 0, 1, 1, 0, 1) - ones: 4, twos: 2 (1, 0, 2, 0, 2, 0, 1, 1, 1, 0) - ones: 4, twos: 2 (1, 1, 0, 0, 1, 1, 0, 2, 0, 2) - ones: 4, twos: 2 (1, 1, 0, 0, 1, 1, 1, 1, 0, 2) - ones: 6, twos: 1 (1, 1, 0, 0, 2, 0, 1, 1, 0, 2) - ones: 4, twos: 2 (1, 1, 0, 0, 2, 0, 1, 1, 1, 1) - ones: 6, twos: 1 (1, 1, 0, 0, 2, 0, 2, 0, 1, 1) - ones: 4, twos: 2 (1, 1, 0, 1, 0, 1, 0, 2, 0, 2) - ones: 4, twos: 2 (1, 1, 0, 1, 0, 1, 1, 1, 0, 2) - ones: 6, twos: 1 (1, 1, 0, 1, 0, 2, 0, 1, 0, 2) - ones: 4, twos: 2 (1, 1, 0, 1, 0, 2, 0, 1, 1, 1) - ones: 6, twos: 1 (1, 1, 0, 1, 0, 2, 0, 2, 0, 1) - ones: 4, twos: 2 (1, 1, 0, 1, 1, 0, 0, 2, 0, 2) - ones: 4, twos: 2 (1, 1, 0, 1, 1, 0, 1, 1, 0, 2) - ones: 6, twos: 1 (1, 1, 0, 1, 1, 0, 2, 0, 0, 2) - ones: 4, twos: 2 (1, 1, 0, 1, 1, 0, 2, 0, 1, 1) - ones: 6, twos: 1 (1, 1, 0, 1, 1, 0, 2, 0, 2, 0) - ones: 4, twos: 2 (1, 1, 0, 1, 1, 1, 0, 1, 0, 2) - ones: 6, twos: 1 (1, 1, 0, 1, 1, 1, 0, 2, 0, 1) - ones: 6, twos: 1 (1, 1, 0, 1, 1, 1, 1, 0, 0, 2) - ones: 6, twos: 1 (1, 1, 0, 1, 1, 1, 1, 0, 2, 0) - ones: 6, twos: 1 (1, 1, 0, 2, 0, 0, 1, 1, 0, 2) - ones: 4, twos: 2 (1, 1, 0, 2, 0, 0, 1, 1, 1, 1) - ones: 6, twos: 1 (1, 1, 0, 2, 0, 0, 2, 0, 1, 1) - ones: 4, twos: 2 (1, 1, 0, 2, 0, 1, 0, 1, 0, 2) - ones: 4, twos: 2 (1, 1, 0, 2, 0, 1, 0, 1, 1, 1) - ones: 6, twos: 1 (1, 1, 0, 2, 0, 1, 0, 2, 0, 1) - ones: 4, twos: 2 (1, 1, 0, 2, 0, 1, 1, 0, 0, 2) - ones: 4, twos: 2 (1, 1, 0, 2, 0, 1, 1, 0, 1, 1) - ones: 6, twos: 1 (1, 1, 0, 2, 0, 1, 1, 0, 2, 0) - ones: 4, twos: 2 (1, 1, 0, 2, 0, 1, 1, 1, 0, 1) - ones: 6, twos: 1 (1, 1, 0, 2, 0, 1, 1, 1, 1, 0) - ones: 6, twos: 1 (1, 1, 0, 2, 0, 2, 0, 0, 1, 1) - ones: 4, twos: 2 (1, 1, 0, 2, 0, 2, 0, 1, 0, 1) - ones: 4, twos: 2 (1, 1, 0, 2, 0, 2, 0, 1, 1, 0) - ones: 4, twos: 2 (1, 1, 1, 0, 0, 1, 0, 2, 0, 2) - ones: 4, twos: 2 (1, 1, 1, 0, 0, 1, 1, 1, 0, 2) - ones: 6, twos: 1 (1, 1, 1, 0, 0, 2, 0, 1, 0, 2) - ones: 4, twos: 2 (1, 1, 1, 0, 0, 2, 0, 1, 1, 1) - ones: 6, twos: 1 (1, 1, 1, 0, 0, 2, 0, 2, 0, 1) - ones: 4, twos: 2 (1, 1, 1, 0, 1, 0, 0, 2, 0, 2) - ones: 4, twos: 2 (1, 1, 1, 0, 1, 0, 1, 1, 0, 2) - ones: 6, twos: 1 (1, 1, 1, 0, 1, 0, 2, 0, 0, 2) - ones: 4, twos: 2 (1, 1, 1, 0, 1, 0, 2, 0, 1, 1) - ones: 6, twos: 1 (1, 1, 1, 0, 1, 0, 2, 0, 2, 0) - ones: 4, twos: 2 (1, 1, 1, 0, 1, 1, 0, 1, 0, 2) - ones: 6, twos: 1 (1, 1, 1, 0, 1, 1, 0, 2, 0, 1) - ones: 6, twos: 1 (1, 1, 1, 0, 1, 1, 1, 0, 0, 2) - ones: 6, twos: 1 (1, 1, 1, 0, 1, 1, 1, 0, 2, 0) - ones: 6, twos: 1 (1, 1, 1, 0, 2, 0, 0, 1, 0, 2) - ones: 4, twos: 2 (1, 1, 1, 0, 2, 0, 0, 1, 1, 1) - ones: 6, twos: 1 (1, 1, 1, 0, 2, 0, 0, 2, 0, 1) - ones: 4, twos: 2 (1, 1, 1, 0, 2, 0, 1, 0, 0, 2) - ones: 4, twos: 2 (1, 1, 1, 0, 2, 0, 1, 0, 1, 1) - ones: 6, twos: 1 (1, 1, 1, 0, 2, 0, 1, 0, 2, 0) - ones: 4, twos: 2 (1, 1, 1, 0, 2, 0, 1, 1, 0, 1) - ones: 6, twos: 1 (1, 1, 1, 0, 2, 0, 1, 1, 1, 0) - ones: 6, twos: 1 (1, 1, 1, 0, 2, 0, 2, 0, 0, 1) - ones: 4, twos: 2 (1, 1, 1, 0, 2, 0, 2, 0, 1, 0) - ones: 4, twos: 2 (1, 1, 1, 1, 0, 0, 0, 2, 0, 2) - ones: 4, twos: 2 (1, 1, 1, 1, 0, 0, 1, 1, 0, 2) - ones: 6, twos: 1 (1, 1, 1, 1, 0, 0, 2, 0, 0, 2) - ones: 4, twos: 2 (1, 1, 1, 1, 0, 0, 2, 0, 1, 1) - ones: 6, twos: 1 (1, 1, 1, 1, 0, 0, 2, 0, 2, 0) - ones: 4, twos: 2 (1, 1, 1, 1, 0, 1, 0, 1, 0, 2) - ones: 6, twos: 1 (1, 1, 1, 1, 0, 1, 0, 2, 0, 1) - ones: 6, twos: 1 (1, 1, 1, 1, 0, 1, 1, 0, 0, 2) - ones: 6, twos: 1 (1, 1, 1, 1, 0, 1, 1, 0, 2, 0) - ones: 6, twos: 1 (1, 1, 1, 1, 0, 2, 0, 0, 0, 2) - ones: 4, twos: 2 (1, 1, 1, 1, 0, 2, 0, 0, 1, 1) - ones: 6, twos: 1 (1, 1, 1, 1, 0, 2, 0, 0, 2, 0) - ones: 4, twos: 2 (1, 1, 1, 1, 0, 2, 0, 1, 0, 1) - ones: 6, twos: 1 (1, 1, 1, 1, 0, 2, 0, 1, 1, 0) - ones: 6, twos: 1 (1, 1, 1, 1, 0, 2, 0, 2, 0, 0) - ones: 4, twos: 2 (1, 1, 1, 1, 1, 0, 0, 1, 0, 2) - ones: 6, twos: 1 (1, 1, 1, 1, 1, 0, 0, 2, 0, 1) - ones: 6, twos: 1 (1, 1, 1, 1, 1, 0, 1, 0, 0, 2) - ones: 6, twos: 1 (1, 1, 1, 1, 1, 0, 1, 0, 2, 0) - ones: 6, twos: 1 (1, 1, 1, 1, 1, 0, 2, 0, 0, 1) - ones: 6, twos: 1 (1, 1, 1, 1, 1, 0, 2, 0, 1, 0) - ones: 6, twos: 1 (1, 1, 1, 1, 1, 1, 0, 0, 0, 2) - ones: 6, twos: 1 (1, 1, 1, 1, 1, 1, 0, 0, 2, 0) - ones: 6, twos: 1 (1, 1, 1, 1, 1, 1, 0, 2, 0, 0) - ones: 6, twos: 1 (2, 0, 0, 0, 1, 1, 1, 1, 0, 2) - ones: 4, twos: 2 (2, 0, 0, 0, 1, 1, 1, 1, 1, 1) - ones: 6, twos: 1 (2, 0, 0, 0, 2, 0, 1, 1, 1, 1) - ones: 4, twos: 2 (2, 0, 0, 1, 0, 1, 1, 1, 0, 2) - ones: 4, twos: 2 (2, 0, 0, 1, 0, 1, 1, 1, 1, 1) - ones: 6, twos: 1 (2, 0, 0, 1, 0, 2, 0, 1, 1, 1) - ones: 4, twos: 2 (2, 0, 0, 1, 1, 0, 1, 1, 0, 2) - ones: 4, twos: 2 (2, 0, 0, 1, 1, 0, 1, 1, 1, 1) - ones: 6, twos: 1 (2, 0, 0, 1, 1, 0, 2, 0, 1, 1) - ones: 4, twos: 2 (2, 0, 0, 1, 1, 1, 0, 1, 0, 2) - ones: 4, twos: 2 (2, 0, 0, 1, 1, 1, 0, 1, 1, 1) - ones: 6, twos: 1 (2, 0, 0, 1, 1, 1, 0, 2, 0, 1) - ones: 4, twos: 2 (2, 0, 0, 1, 1, 1, 1, 0, 0, 2) - ones: 4, twos: 2 (2, 0, 0, 1, 1, 1, 1, 0, 1, 1) - ones: 6, twos: 1 (2, 0, 0, 1, 1, 1, 1, 0, 2, 0) - ones: 4, twos: 2 (2, 0, 0, 1, 1, 1, 1, 1, 0, 1) - ones: 6, twos: 1 (2, 0, 0, 1, 1, 1, 1, 1, 1, 0) - ones: 6, twos: 1 (2, 0, 0, 2, 0, 0, 1, 1, 1, 1) - ones: 4, twos: 2 (2, 0, 0, 2, 0, 1, 0, 1, 1, 1) - ones: 4, twos: 2 (2, 0, 0, 2, 0, 1, 1, 0, 1, 1) - ones: 4, twos: 2 (2, 0, 0, 2, 0, 1, 1, 1, 0, 1) - ones: 4, twos: 2 (2, 0, 0, 2, 0, 1, 1, 1, 1, 0) - ones: 4, twos: 2 (2, 0, 1, 0, 0, 1, 1, 1, 0, 2) - ones: 4, twos: 2 (2, 0, 1, 0, 0, 1, 1, 1, 1, 1) - ones: 6, twos: 1 (2, 0, 1, 0, 0, 2, 0, 1, 1, 1) - ones: 4, twos: 2 (2, 0, 1, 0, 1, 0, 1, 1, 0, 2) - ones: 4, twos: 2 (2, 0, 1, 0, 1, 0, 1, 1, 1, 1) - ones: 6, twos: 1 (2, 0, 1, 0, 1, 0, 2, 0, 1, 1) - ones: 4, twos: 2 (2, 0, 1, 0, 1, 1, 0, 1, 0, 2) - ones: 4, twos: 2 (2, 0, 1, 0, 1, 1, 0, 1, 1, 1) - ones: 6, twos: 1 (2, 0, 1, 0, 1, 1, 0, 2, 0, 1) - ones: 4, twos: 2 (2, 0, 1, 0, 1, 1, 1, 0, 0, 2) - ones: 4, twos: 2 (2, 0, 1, 0, 1, 1, 1, 0, 1, 1) - ones: 6, twos: 1 (2, 0, 1, 0, 1, 1, 1, 0, 2, 0) - ones: 4, twos: 2 (2, 0, 1, 0, 1, 1, 1, 1, 0, 1) - ones: 6, twos: 1 (2, 0, 1, 0, 1, 1, 1, 1, 1, 0) - ones: 6, twos: 1 (2, 0, 1, 0, 2, 0, 0, 1, 1, 1) - ones: 4, twos: 2 (2, 0, 1, 0, 2, 0, 1, 0, 1, 1) - ones: 4, twos: 2 (2, 0, 1, 0, 2, 0, 1, 1, 0, 1) - ones: 4, twos: 2 (2, 0, 1, 0, 2, 0, 1, 1, 1, 0) - ones: 4, twos: 2 (2, 0, 1, 1, 0, 0, 1, 1, 0, 2) - ones: 4, twos: 2 (2, 0, 1, 1, 0, 0, 1, 1, 1, 1) - ones: 6, twos: 1 (2, 0, 1, 1, 0, 0, 2, 0, 1, 1) - ones: 4, twos: 2 (2, 0, 1, 1, 0, 1, 0, 1, 0, 2) - ones: 4, twos: 2 (2, 0, 1, 1, 0, 1, 0, 1, 1, 1) - ones: 6, twos: 1 (2, 0, 1, 1, 0, 1, 0, 2, 0, 1) - ones: 4, twos: 2 (2, 0, 1, 1, 0, 1, 1, 0, 0, 2) - ones: 4, twos: 2 (2, 0, 1, 1, 0, 1, 1, 0, 1, 1) - ones: 6, twos: 1 (2, 0, 1, 1, 0, 1, 1, 0, 2, 0) - ones: 4, twos: 2 (2, 0, 1, 1, 0, 1, 1, 1, 0, 1) - ones: 6, twos: 1 (2, 0, 1, 1, 0, 1, 1, 1, 1, 0) - ones: 6, twos: 1 (2, 0, 1, 1, 0, 2, 0, 0, 1, 1) - ones: 4, twos: 2 (2, 0, 1, 1, 0, 2, 0, 1, 0, 1) - ones: 4, twos: 2 (2, 0, 1, 1, 0, 2, 0, 1, 1, 0) - ones: 4, twos: 2 (2, 0, 1, 1, 1, 0, 0, 1, 0, 2) - ones: 4, twos: 2 (2, 0, 1, 1, 1, 0, 0, 1, 1, 1) - ones: 6, twos: 1 (2, 0, 1, 1, 1, 0, 0, 2, 0, 1) - ones: 4, twos: 2 (2, 0, 1, 1, 1, 0, 1, 0, 0, 2) - ones: 4, twos: 2 (2, 0, 1, 1, 1, 0, 1, 0, 1, 1) - ones: 6, twos: 1 (2, 0, 1, 1, 1, 0, 1, 0, 2, 0) - ones: 4, twos: 2 (2, 0, 1, 1, 1, 0, 1, 1, 0, 1) - ones: 6, twos: 1 (2, 0, 1, 1, 1, 0, 1, 1, 1, 0) - ones: 6, twos: 1 (2, 0, 1, 1, 1, 0, 2, 0, 0, 1) - ones: 4, twos: 2 (2, 0, 1, 1, 1, 0, 2, 0, 1, 0) - ones: 4, twos: 2 (2, 0, 1, 1, 1, 1, 0, 0, 0, 2) - ones: 4, twos: 2 (2, 0, 1, 1, 1, 1, 0, 0, 1, 1) - ones: 6, twos: 1 (2, 0, 1, 1, 1, 1, 0, 0, 2, 0) - ones: 4, twos: 2 (2, 0, 1, 1, 1, 1, 0, 1, 0, 1) - ones: 6, twos: 1 (2, 0, 1, 1, 1, 1, 0, 1, 1, 0) - ones: 6, twos: 1 (2, 0, 1, 1, 1, 1, 0, 2, 0, 0) - ones: 4, twos: 2 (2, 0, 1, 1, 1, 1, 1, 0, 0, 1) - ones: 6, twos: 1 (2, 0, 1, 1, 1, 1, 1, 0, 1, 0) - ones: 6, twos: 1 (2, 0, 1, 1, 1, 1, 1, 1, 0, 0) - ones: 6, twos: 1 (2, 0, 2, 0, 0, 0, 1, 1, 1, 1) - ones: 4, twos: 2 (2, 0, 2, 0, 0, 1, 0, 1, 1, 1) - ones: 4, twos: 2 (2, 0, 2, 0, 0, 1, 1, 0, 1, 1) - ones: 4, twos: 2 (2, 0, 2, 0, 0, 1, 1, 1, 0, 1) - ones: 4, twos: 2 (2, 0, 2, 0, 0, 1, 1, 1, 1, 0) - ones: 4, twos: 2 (2, 0, 2, 0, 1, 0, 0, 1, 1, 1) - ones: 4, twos: 2 (2, 0, 2, 0, 1, 0, 1, 0, 1, 1) - ones: 4, twos: 2 (2, 0, 2, 0, 1, 0, 1, 1, 0, 1) - ones: 4, twos: 2 (2, 0, 2, 0, 1, 0, 1, 1, 1, 0) - ones: 4, twos: 2 (2, 0, 2, 0, 1, 1, 0, 0, 1, 1) - ones: 4, twos: 2 (2, 0, 2, 0, 1, 1, 0, 1, 0, 1) - ones: 4, twos: 2 (2, 0, 2, 0, 1, 1, 0, 1, 1, 0) - ones: 4, twos: 2 (2, 0, 2, 0, 1, 1, 1, 0, 0, 1) - ones: 4, twos: 2 (2, 0, 2, 0, 1, 1, 1, 0, 1, 0) - ones: 4, twos: 2 (2, 0, 2, 0, 1, 1, 1, 1, 0, 0) - ones: 4, twos: 2 [/PYTHON OUTPUT]
Reasoning Process

[THOUGHT] The Python code enumerated all valid configurations of 10 boxes with 8 balls where:

  1. Total balls = 8
  2. Each box has 0, 1, or 2 balls
  3. Condition A: The number of boxes with exactly 1 ball is either 4 or 6
  4. Condition B: Any box with 2 balls has empty (0 ball) neighbors on both sides

The code found 262 valid configurations that satisfy all these conditions. [/THOUGHT]

\boxed{262}

Final Model Output
262

Comparative Analysis (P39) (Average across runs)

Avg Time Taken

Solar-Pro-2 (31B)(high)
120.6
Gemini-3-Pro-Preview
93.8
Deepseek-V3.2
80.4
Grok-4.1-fast
52.3
Llama-VARCO-8B-Instruct
39.2
HCX-007(high)
34.1
A.X-4.0 (72B)
17.5
Claude-Opus-4.5
16.5
GPT-5.1 (high)
15.3
EXAONE-4.0.1-32B (high)
6.8

Avg Token Usage

Solar-Pro-2 (31B)(high)
16125.0
Claude-Opus-4.5
14865.0
Gemini-3-Pro-Preview
10966.0
Grok-4.1-fast
7391.0
Deepseek-V3.2
7336.0
EXAONE-4.0.1-32B (high)
4443.0
HCX-007(high)
4160.0
A.X-4.0 (72B)
3832.0
GPT-5.1 (high)
2977.0
Llama-VARCO-8B-Instruct
806.0