"""Sharper result for D=5: at least 8 pairs MUST be left uncovered.

For ANY 5 balanced partitions (boats of size 6):
  sum_{x<y} C(a_xy, 2) = sum_{c<c'} D_cc'  >= 10 * 8 = 80        (forced)
  sum_{x<y} a_xy = 5*60 = 300                                    (always)
  a_xy in {0,..,5}.
Let C = #covered pairs (a>=1). Write a=1+e on covered pairs, sum e = 300 - C.
Constraint: sum e(e+1)/2 >= 80  =>  sum e^2 >= 160-(300-C) = C-140.
Since e<=4: sum e^2 <= 4 sum e = 4(300-C).
  => C-140 <= 4(300-C) => 5C <= 1340 => C <= 268.
So >= 276-268 = 8 pairs are uncovered with 5 days.
"""
from math import comb

DAYS = 5
total_agreements = DAYS * 60           # sum of a_xy
forced_double = 8 * comb(DAYS, 2)      # lower bound on sum C(a,2)
print(f"sum a_xy           = {total_agreements}")
print(f"forced sum C(a,2) >= {forced_double}")

# solve for max C
# C - 140 <= 4(300 - C)
best_C = None
for C in range(276, 0, -1):
    e_sum = total_agreements - C            # = sum e
    need = forced_double * 2 - total_agreements + C   # sum e^2 >= 160 - (300-C)? recompute
    # constraint: sum e^2 >= 2*forced_double - e_sum   (since sum e(e+1)/2>=fd => sum e^2+sum e>=2fd)
    lhs_min_needed = 2 * forced_double - e_sum
    max_possible_e2 = 4 * e_sum            # e<=4
    if max_possible_e2 >= lhs_min_needed:
        best_C = C
        break
print(f"max coverable pairs C = {best_C}")
print(f"=> at least {276 - best_C} pairs uncovered with {DAYS} days (full cover impossible)")