bulgarian_solitaire_riddle.py

from itertools import combinations
from functools import reduce
from pprint import pprint as pr
  
def all_arrangements(k):
	m=k*(k+1)/2
	m_bits_on=set([tuple(reduce(lambda x,y:x[:y]+[1]+x[y+1:],c,[0]*(2*m+1))) for c in combinations(range(2*m+1),m)])
	return set([tuple(sorted(filter(lambda i:i>0,reduce(lambda x,y: x+[y] if y==0 else x[:-1]+[x[-1]+1,],p,[0])))) for p in m_bits_on])

def move(t,n=1):
	if n>1:
		return move(move(t),n-1)
	return tuple(sorted([x-1 for x in t if x>1]+[len(t)]))

def verify(f,a):
	arrangements=sorted(map(lambda x: (f(x),f(x)-f(move(x)),x),a))
	pr(arrangements)
	return all([v[1]<0 for v in arrangements]) or all([v[1]>0 for v in arrangements])

if __name__=='__main__':
	a = all_arrangements(3)
	#f is proportional to the potential energy of the young diagram rotated by 45 degrees
	#see: https://arxiv.org/pdf/1503.00885.pdf
	f = lambda x: sum([(len(x)-i)*t+(t+1)*t/2.0 for i,t in enumerate(x)])
	verify(f,a)