Buggy python submission (verdict: Time Limit Exceeded):
h, w, k = map(int, input().split())

a = [[0 for i in range(w)] for j in range(h)]

for i in range(h):
    s = input()
    for j in range(w):
        a[i][j] = int(s[j])

best_result = h * w + 1000

for m in range(2 ** (h - 1)):
    mask = m
    num_splits = 0
    splits = []
    i = 0
    while mask > 0:
        if (mask % 2) == 1:
            num_splits += 1
            splits.append(i)
        i += 1
        mask = mask // 2

    #print('mask = ', m, 'splits = ', splits, 'num_splits =', num_splits)

    counts = [0 for j in range(num_splits + 1)]
    current_counts = [0 for j in range(num_splits + 1)]
    result = num_splits

    for j in range(w):
        if result >= best_result:
            break
        split_index = 0
        success = True
        current_counts[0] = 0

        for i in range(h):
            if a[i][j] == 1:
                counts[split_index] += 1
                if counts[split_index] > k:
                    success = False

            current_counts[split_index] += a[i][j]

            if split_index < num_splits and i == splits[split_index]:
                split_index += 1
                current_counts[split_index] = 0

        if not success:
            result += 1
            #print('not success j = ', j, 'result = ', result, 'counts = ', counts)
            for x in range(num_splits + 1):
                counts[x] = current_counts[x]
            

    best_result = min(best_result, result)

print(best_result)

Fixed submission:
h, w, k = map(int, input().split())

a = [[1 for i in range(w)] for j in range(h)]

for i in range(h):
    s = input()
    for j in range(w):
        a[i][j] = int(s[j])

best_result = h * w + 1000

for m in range(2 ** (h - 1)):
    mask = m
    num_splits = 0
    splits = []
    i = 0
    while mask > 0:
        if (mask % 2) == 1:
            num_splits += 1
            splits.append(i)
        i += 1
        mask = mask // 2

    #print('mask = ', m, 'splits = ', splits, 'num_splits =', num_splits)

    counts = [0 for j in range(num_splits + 1)]
    current_counts = [0 for j in range(num_splits + 1)]
    result = num_splits

    for j in range(w):
        if result >= best_result:
            break
        split_index = 0
        success = True
        current_counts[0] = 0

        for i in range(h):
            if a[i][j] == 1:
                counts[split_index] += 1
                if counts[split_index] > k:
                    success = False

            current_counts[split_index] += a[i][j]

            if split_index < num_splits and i == splits[split_index]:
                split_index += 1
                current_counts[split_index] = 0

        if not success:
            result += 1
            #print('not success j = ', j, 'result = ', result, 'counts = ', counts)
            fail = False
            for x in range(num_splits + 1):
                if current_counts[x] > k:
                    fail = True
                    break

                counts[x] = current_counts[x]
            if fail: 
                result = best_result + 1
                break
            

    best_result = min(best_result, result)

print(best_result)