インテジャーズ

INTEGERS

数、特に整数に関する記事。

8951

整数 整数-8951

8951は素数ですが、23 \ (=8+9+5+1)の分割を

\begin{align}23&=22+1 \\ &=21+2\\ &=21+1+1\\ &=20+3\\ &=20+2+1\\ &=20+1+1+1\\ &=19+4\\ &=19+3+1\\ &=19+2+2\\ &=19+2+1+1\\ &=19+1+1+1+1\\ &=18+5\\ &=18+4+1\\ &=18+3+2\\ &=18+3+ 1+ 1\\ &=18+2+ 2+ 1\\ &=18+2+ 1+ 1+ 1\\ &=18+1+ 1+ 1+ 1+ 1\\ &=17+6 \\ &=17+5+ 1 \\ &=17+4+ 2 \\ &=17+4+ 1+ 1 \\ &=17+3+ 3 \\ &=17+3+ 2+ 1 \\ &=17+3+ 1+ 1+ 1 \\ &=17+2+ 2+ 2 \\ &=17+2+ 2+ 1+ 1 \\ &=17+2+ 1+ 1+ 1+ 1 \\ &=17+1+ 1+ 1+ 1+ 1+ 1 \\ &=16+7 \\ &=16+6+ 1 \\ &=16+5+ 2 \\ &=16+5+ 1+ 1 \\ &=16+4+ 3 \\ &=16+4+ 2+ 1 \\ &=16+4+ 1+ 1+ 1 \\ &=16+3+ 3+ 1 \\ &=16+3+ 2+ 2 \\ &=16+3+ 2+ 1+ 1 \\ &=16+3+ 1+ 1+ 1+ 1 \\ &=16+2+ 2+ 2+ 1 \\ &=16+2+ 2+ 1+ 1+ 1 \\ &=16+2+ 1+ 1+ 1+ 1+ 1 \\ &=16+1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=15+8 \\ &=15+7+ 1 \\ &=15+6+ 2 \\ &=15+6+ 1+ 1 \\ &=15+5+ 3 \\ &=15+5+ 2+ 1 \\ &=15+5+ 1+ 1+ 1 \\ &=15+4+ 4 \\ &=15+4+ 3+ 1 \\ &=15+4+ 2+ 2 \\ &=15+4+ 2+ 1+ 1 \\ &=15+4+ 1+ 1+ 1+ 1 \\ &=15+3+ 3+ 2 \\ &=15+3+ 3+ 1+ 1 \\ &=15+3+ 2+ 2+ 1 \\ &=15+3+ 2+ 1+ 1+ 1\\ &=15+3+ 1+ 1+ 1+ 1+ 1 \\ &=15+2+ 2+ 2+ 2 \\ &=15+2+ 2+ 2+ 1+ 1 \\ &=15+2+ 2+ 1+ 1+ 1+ 1 \\ &=15+2+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=15+1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=14+9 \\ &=14+8+ 1 \\ &=14+7+ 2 \\ &=14+7+ 1+ 1 \\ &=14+6+ 3 \\ &=14+6+ 2+ 1 \\ &=14+6+ 1+ 1+ 1 \\ &=14+5+ 4 \\ &=14+5+ 3+ 1 \\ &=14+5+ 2+ 2 \\ &=14+5+ 2+ 1+ 1 \\ &=14+5+ 1+ 1+ 1+ 1 \\ &=14+4+ 4+ 1 \\ &=14+4+ 3+ 2 \\ &=14+4+ 3+ 1+ 1 \\ &=14+4+ 2+ 2+ 1 \\ &=14+4+ 2+ 1+ 1+ 1 \\ &=14+4+ 1 +1+ 1+ 1+ 1 \\ &=14+3+ 3+ 3 \\ &=14+3+ 3+ 2+ 1 \\ &=14+3+ 3+ 1+ 1+ 1 \\ &=14+3+ 2+ 2+ 2 \\ &=14+3+ 2+ 2+ 1+ 1 \\ &=14+3+ 2+ 1+ 1+ 1+ 1 \\ &=14+3 +1 +1+ 1+ 1+ 1+ 1 \\ &=14+2+ 2+ 2 +2+ 1 \\ &=14+2+ 2+ 2+ 1+ 1+ 1 \\ &=14+2+ 2+ 1+ 1+ 1+ 1+ 1 \\ &=14+2+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=14+1+ 1+ 1+ 1+ 1+ 1 +1 +1+ 1 \\ &=13+10 \\ &=13+9+ 1 \\ &=13+8+ 2 \\ &=13+8+ 1+ 1 \\ &=13+7+ 3 \\ &=13+7+ 2+ 1 \\ &=13+7+ 1+ 1+ 1 \\ &=13+6+ 4 \\ &=13+6+ 3+ 1 \\ &=13+6+ 2+ 2 \\ &=13+6+ 2+ 1+ 1 \\ &=13+6+ 1+ 1+ 1+ 1 \\ &=13+5+ 5 \\ &=13+5+ 4+ 1 \\ &=13+5+ 3+ 2 \\ &=13+5+ 3+ 1+ 1 \\ &=13+5+ 2+ 2+ 1 \\ &=13+5+ 2+ 1+ 1+ 1 \\ &=13+5+ 1+ 1+ 1+ 1+ 1 \\ &=13+4+ 4+ 2 \\ &=13+4+ 4+ 1+ 1 \\ &=13+4+ 3+ 3 \\ &=13+4+ 3+ 2+ 1 \\ &=13+4+ 3+ 1+ 1+ 1 \\ &=13+4+ 2+ 2+ 2 \\ &=13+4+ 2+ 2+ 1+ 1 \\ &=13+4+ 2+ 1+ 1+ 1+ 1 \\ &=13+4+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=13+3+ 3+ 3+ 1 \\ &=13+3+ 3+ 2+ 2 \\ &=13+3+ 3+ 2+ 1+ 1 \\ &=13+3+ 3+ 1+ 1+ 1+ 1 \\ &=13+3+ 2+ 2+ 2+ 1 \\ &=13+3+ 2+ 2+ 1+ 1+ 1 \\ &=13+3+ 2+ 1+ 1+ 1+ 1+ 1 \\ &=13+3+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=13+2+ 2+ 2+ 2+ 2 \\ &=13+2+ 2+ 2+ 2+ 1+ 1 \\ &=13+2+ 2+ 2+ 1+ 1+ 1+ 1 \\ &=13+2+ 2+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=13+2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=13+1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=12+11 \\ &=12+10+ 1 \\ &=12+9+ 2 \\ &=12+9+ 1+ 1 \\ &=12+8+ 3 \\ &=12+8+ 2+ 1 \\ &=12+8+ 1+ 1+ 1 \\ &=12+7+ 4 \\ &=12+7+ 3+ 1 \\ &= 12+7+ 2+ 2 \\ &=12+7+ 2+ 1+ 1 \\ &=12+7+ 1+ 1+ 1+ 1 \\ &=12+6+ 5 \\ &=12+6+ 4+ 1 \\ &=12+6+ 3+ 2 \\ &=12+6+ 3+ 1+ 1 \\ &=12+6+ 2+ 2+ 1 \\ &=12+6+ 2+ 1+ 1+ 1 \\ &=12+6+ 1+ 1+ 1+ 1+ 1 \\ &=12+5+ 5+ 1 \\ &=12+5+ 4+ 2 \\ &=12+5+ 4+ 1+ 1 \\ &=12+5+ 3+ 3 \\ &=12+5+ 3+ 2+ 1 \\ &=12+5+ 3+ 1+ 1+ 1 \\ &=12+5+ 2+ 2+ 2 \\ &=12+5+ 2+ 2+ 1+ 1 \\ &=12+5+ 2+ 1+ 1+ 1+ 1 \\ &=12+5+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=12+4+ 4+ 3 \\ &=12+4+ 4+ 2+ 1 \\ &=12+4+ 4+ 1+ 1+ 1 \\ &=12+4+ 3+ 3+ 1 \\ &=12+4+ 3+ 2+ 2 \\ &=12+4+ 3+ 2+ 1+ 1 \\ &=12+4+ 3+ 1+ 1+ 1+ 1 \\ &=12+4+ 2+ 2+ 2+ 1 \\ &=12+4+ 2+ 2+ 1+ 1+ 1 \\ &= 12+4+ 2+ 1+ 1+ 1+ 1+ 1 \\ &= 12+4+ 1+ 1+ 1+ 1+ 1+ 1+ 1\\ &= 12+3+ 3+ 3+ 2 \\ &= 12+3+ 3+ 3+ 1+ 1\\ &= 12+3+ 3+ 2+ 2+ 1 \\ &= 12+3+ 3+ 2+ 1+ 1+ 1\\ &= 12+3+ 3+ 1+ 1 +1 +1 +1\\ &= 12+3+ 2+ 2 +2+ 2 \\ &= 12+3+ 2+ 2+ 2+ 1+ 1\\ &= 12+3+ 2+ 2+ 1+ 1+ 1+ 1\\ &= 12+3+ 2+ 1+ 1+ 1+ 1+ 1+ 1\\ &= 12+3+1+ 1+ 1+ 1+ 1+ 1+ 1+ 1\\ &= 12+2+ 2+ 2+ 2+ 2+ 1 \\ &= 12+2+ 2+ 2+ 2+ 1+ 1+ 1+ \\ &= 12+2 +2+ 2+ 1+ 1+ 1+ 1+ 1\\ &= 12+2+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &= 12+2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1\\ &=12+1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=11+11+ 1 \\ &=11+10+ 2 \\ &=11+10+ 1+ 1 \\ &=11+9+ 3 \\ &=11+9+ 2+ 1 \\ &=11+9+ 1+ 1+ 1 \\ &=11+8+ 4 \\ &=11+8+ 3+ 1 \\ &=11+8+ 2+ 2 \\ &=11+8+ 2+ 1+ 1 \\ &=11+8+ 1+ 1+ 1+ 1 \\ &=11+7+ 5 \\ &=11+7+ 4+ 1 \\ &=11+7+ 3+ 2 \\ &=11+7+ 3+ 1+ 1 \\ &=11+7+ 2+ 2+ 1 \\ &=11+7+ 2+ 1+ 1+ 1 \\ &=11+7+ 1+ 1+ 1+ 1+ 1 \\ &=11+6+ 6 \\ &=11+6+ 5+ 1 \\ &=11+6+ 4+ 2 \\ &=11+6+ 4+ 1+ 1 \\ &=11+6+ 3+ 3+ \\ &=11+6+ 3+ 2+ 1+ \\ &=11+6+ 3+ 1+ 1+ 1 \\ &=11+6+ 2+ 2+ 2 \\ &=11+6+ 2+ 2+ 1+ 1 \\ &=11+6+ 2+ 1+ 1+ 1+ 1 \\ &=11+6+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=11+5+ 5+ 2 \\ &=11+5+ 5+ 1+ 1 \\ &=11+5+ 4+ 3 \\ &=11+5+ 4+ 2+ 1 \\ &=11+5+ 4+ 1+ 1+ 1 \\ &=11+5 +3 +3+ 1 \\ &=11+5+ 3+ 2+ 2 \\ &=11+5+ 3+ 2+ 1+ 1 \\ &=11+5+ 3+ 1+ 1+ 1+ 1 \\ &=11+5+ 2+ 2+ 2+ 1 \\ &=11+5+ 2+ 2+ 1+ 1+ 1 \\ &=11+5+ 2+ 1+ 1+ 1+ 1+ 1 \\ &=11+5+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=11+4+ 4+ 4 \\ &=11+4+ 4+ 3+ 1 \\ &=11+4+ 4+ 2+ 2 \\ &=11+4+ 4+ 2+ 1+ 1 \\ &=11+4+ 4+ 1+ 1+ 1+ 1 \\ &=11+4+ 3+ 3+ 2 \\ &=11+4+ 3+ 3+ 1+ 1 \\ &=11+4+ 3+ 2+ 2+ 1 \\ &=11+4+ 3+ 2+ 1+ 1+ 1 \\ &=11+4+ 3+ 1+ 1+ 1+ 1+ 1 \\ &=11+4+ 2+ 2+ 2+ 2 \\ &=11+4+ 2+ 2+ 2+ 1+ 1 \\ &=11+4+ 2+ 2+ 1+ 1+ 1+ 1 \\ &=11+4+ 2+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=11+4+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=11+3+ 3+ 3+ 3 \\ &=11+3+ 3+ 3+ 2+ 1 \\ &=11+3+ 3+ 3+ 1+ 1+ 1 \\ &=11+3+ 3+ 2+ 2+ 2 \\ &=11+3+ 3+ 2+ 2+ 1+ 1 \\ &=11+3+ 3+ 2+ 1+ 1+ 1+ 1 \\ &=11+3+ 3+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=11+3+ 2+ 2+ 2+ 2+ 1 \\ &= 11+3+ 2+ 2+ 2+ 1+ 1+ 1 \\ &=11+3+ 2+ 2+ 1+1+ 1+ 1+ 1 \\ &=11+3+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=11+3+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=11+2+ 2+ 2+ 2+ 2+ 2 \\ &=11+2 +2+ 2+ 2+ 2+ 1+ 1 \\ &=11+2+ 2+ 2+ 2+ 1+ 1+ 1+ 1 \\ &=11+2+ 2+ 2+ 1 +1+ 1+ 1+ 1+ 1 \\ &=11+2+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=11+2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=11+1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=10+10+ 3 \\ &=10+10+ 2+ 1 \\ &=10+10+ 1+ 1+ 1 \\ &=10+9+ 4 \\ &=10+9+ 3+ 1 \\ &=10+9+ 2+ 2 \\ &=10+9+ 2+ 1+ 1 \\ &=10+9+ 1+ 1+ 1+ 1 \\ &=10+8+ 5 \\ &=10+8+ 4+ 1 \\ &=10+8+ 3+ 2 \\ &=10+8+ 3+ 1+ 1 \\ &=10+8+ 2+ 2+ 1 \\ &=10+8+ 2+ 1+ 1+ 1 \\ &=10+8+ 1+ 1+ 1+ 1+ 1 \\ &=10+7+ 6 \\ &=10+7+ 5+ 1 \\ &=10+7+ 4+ 2 \\ &=10+7+ 4+ 1+ 1 \\ &=10+7+ 3+ 3 \\ &=10+7+ 3+ 2+ 1 \\ &=10+7+ 3+ 1+ 1+ 1 \\ &=10+7+ 2+ 2+ 2 \\ &=10+7+ 2+ 2+ 1+ 1 \\ &=10+7+ 2+ 1+ 1+ 1+ 1 \\ &=10+7+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=10+6+ 6+ 1 \\ &=10+6+ 5+ 2 \\ &=10+6+ 5+ 1+ 1 \\ &=10+6+ 4+ 3 \\ &= 10+6+ 4+ 2+ 1 \\ &=10+6+ 4+ 1+ 1+ 1 \\ &=10+6+ 3+ 3+ 1 \\ &=10+6+ 3+ 2+ 2 \\ &=10+6+ 3+ 2+ 1+ 1 \\ &=10+6+ 3+ 1+ 1+ 1+ 1 \\ &=10+6+ 2+ 2+ 2+ 1 \\ &=10+6+ 2+ 2+ 1+ 1+ 1 \\ &=10+6+ 2+ 1+ 1+ 1+ 1+ 1 \\ &=10+6+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=10+5+ 5+ 3 \\ &=10+5+ 5+ 2+ 1 \\ &=10+5+ 5+ 1+ 1+ 1 \\ &= 10+5+ 4+ 4 \\ &=10+5+ 4+ 3+ 1 \\ &=10+5+ 4+ 2+ 2 \\ &=10+5+ 4+ 2+ 1+ 1 \\ &=10+5+ 4+ 1+ 1+ 1+ 1 \\ &=10+5+ 3+ 3+ 2 \\ &=10+5+ 3+ 3+ 1+ 1 \\ &=10+5+ 3+ 2+ 2+ 1 \\ &=10+5+ 3+ 2+ 1+ 1+ 1 \\ &=10+5+ 3+ 1+ 1+ 1+ 1+ 1 \\ &= 10+5+ 2+ 2+ 2+ 2 \\ &=10+5+ 2+ 2+ 2+ 1+ 1 \\ &=10+5 +2+ 2+ 1+ 1+ 1+ 1 \\ &=10+5+ 2+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=10+5+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=10+4+ 4+ 4+ 1 \\ &=10+4+ 4+ 3+ 2 \\ &=10+4+ 4+ 3+ 1+ 1 \\ &=10+4 +4 +2+ 2+ 1 \\ &=10+4+ 4+ 2+ 1+ 1+ 1 \\ &=10+4+ 4+ 1+ 1+ 1+ 1+ 1 \\ &=10+4+ 3+ 3+ 3 \\ &=10+4+ 3+ 3+ 2+ 1 \\ &=10+4 +3 +3+ 1+ 1+ 1 \\ &=10+4+ 3+ 2+ 2+ 2 \\ &=10+4+ 3+ 2+ 2+ 1+ 1 \\ &=10+4+ 3+ 2+ 1+ 1+ 1+ 1 \\ &=10+4+ 3+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=10+4+ 2+ 2+ 2+ 2+ 1 \\ &=10+4+ 2+ 2+ 2+ 1+ 1+ 1 \\ &= 10+4+ 2+ 2+ 1+ 1+ 1+ 1+ 1 \\ &=10+4+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=10+4 +1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=10+3+ 3+ 3+ 3+ 1 \\ &=10+3+ 3+ 3+ 2+ 2 \\ &=10+3+ 3+ 3+ 2+ 1+ 1 \\ &=10+3+ 3+ 3+ 1+ 1+ 1+ 1 \\ &=10+3+ 3+ 2+ 2+ 2+ 1 \\ &=10+3+ 3+ 2+ 2+ 1+ 1+ 1 \\ &= 10+3+ 3+ 2+ 1+ 1+ 1+ 1+ 1 \\ &=10+3+ 3+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &= 10+3+ 2+ 2+ 2+ 2+ 2 \\ &=10+3+ 2+ 2+ 2+ 2+ 1+ 1 \\ &=10+3+ 2+ 2+ 2+ 1+ 1+ 1+ 1 \\ &=10+3+ 2+ 2+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=10+3+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=10+3+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=10+2+ 2+ 2+ 2+ 2+ 2+ 1 \\ &=10+2+ 2+ 2+ 2+ 2+ 1+ 1+ 1 \\ &=10+2+ 2+ 2+ 2+ 1+ 1+ 1+ 1+ 1 \\ &=10+2+ 2+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=10+2+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=10+2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=10+1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=9+9+ 5 \\ &=9+9+ 4+ 1 \\ &= 9+9+ 3+ 2 \\ &=9+9+ 3+ 1+ 1 \\ &= 9+9+ 2+ 2+ 1 \\ &=9+9+ 2+ 1+ 1+ 1 \\ &=9+9+ 1+ 1+ 1+ 1+ 1 \\ &=9+8+ 6 \\ &=9+8+ 5+ 1 \\ &=9+8+ 4+ 2 \\ &=9+8+ 4+ 1+ 1 \\ &=9+8+ 3+ 3 \\ &=9+8+ 3+ 2+ 1 \\ &=9+8+ 3+ 1+ 1+ 1 \\ &=9+8+ 2+ 2+ 2 \\ &=9+8+ 2+ 2+ 1+ 1 \\ &=9+8+ 2+ 1+ 1+ 1+ 1 \\ &=9+8+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=9+7+ 7 \\ &=9+7+ 6+ 1 \\ &=9+7+ 5+ 2 \\ &=9+7+ 5+ 1+ 1 \\ &=9+7 +4+ 3 \\ &=9+7+ 4+ 2+ 1 \\ &=9+7+ 4+ 1+ 1+ 1 \\ &=9+7+ 3+ 3+ 1 \\ &=9+7+ 3+ 2+ 2 \\ &=9+7+ 3+ 2+ 1+ 1 \\ &=9+7+ 3+ 1+ 1+ 1+ 1 \\ &=9+7+ 2+ 2+ 2+ 1 \\ &=9+7+ 2+ 2+ 1+ 1+ 1 \\ &=9+7+ 2+ 1+ 1+ 1+ 1+ 1 \\ &=9+7+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=9+6+ 6+ 2 \\ &=9+6+ 6+ 1+ 1 \\ &=9+6+ 5+ 3 \\ &=9+6+ 5+ 2+ 1 \\ &=9+6+ 5+ 1+ 1+ 1 \\ &=9+6+ 4+ 4 \\ &=9+6+ 4+ 3+ 1 \\ &=9+6+ 4+ 2+ 2 \\ &=9+6+ 4+ 2+ 1+ 1 \\ &=9+6+ 4+ 1+ 1+ 1+ 1 \\ &=9+6+ 3+ 3+ 2 \\ &=9+6+ 3+ 3+ 1+ 1 \\ &=9+6+ 3+ 2+ 2+ 1 \\ &=9+6+ 3+ 2+ 1+ 1+ 1 \\ &=9+6+ 3+ 1+ 1+ 1+ 1+ 1 \\ &=9+6+ 2+ 2+ 2+ 2 \\ &=9+6+ 2+ 2+ 2+ 1+ 1 \\ &=9+6+ 2+ 2+ 1+ 1+ 1+ 1 \\ &=9+6+ 2+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=9+6+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=9+5+ 5+ 4 \\ &=9+5+ 5+ 3+ 1 \\ &=9+5+ 5+ 2+ 2 \\ &=9+5+ 5+ 2+ 1+ 1 \\ &=9+5+ 5+ 1+ 1+ 1+ 1 \\ &=9+5+ 4+ 4+ 1 \\ &=9+5+ 4+ 3+ 2 \\ &= 9+5+ 4+ 3+ 1+ 1 \\ &=9+5+ 4+ 2+ 2+ 1 \\ &=9+5+ 4+ 2+ 1+ 1+ 1 \\ &=9+5+ 4+ 1+ 1+ 1+ 1+ 1 \\ &=9+5+ 3+ 3+ 3 \\ &=9+5+ 3+ 3+ 2+ 1 \\ &=9+5+ 3+ 3+ 1+ 1+ 1 \\ &=9+5+ 3+ 2+ 2+ 2 \\ &= 9+5+ 3+ 2+ 2+ 1+ 1 \\ &=9+5+ 3+ 2+ 1+ 1+ 1+ 1 \\ &=9+5+ 3+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=9+5+ 2+ 2+ 2+ 2+ 1 \\ &=9+5+ 2+ 2+ 2+ 1+ 1+ 1 \\ &=9+5+ 2+ 2+ 1+ 1+ 1+ 1+ 1 \\ &= 9+5 +2 +1 +1+ 1+ 1+ 1+ 1+ 1 \\ &=9+5+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=9+4+ 4+ 4+ 2 \\ &=9+4+ 4+ 4+ 1+ 1 \\ &=9+4+ 4+ 3+ 3 \\ &=9+4+ 4+ 3+ 2+ 1 \\ &=9+4+ 4+ 3+ 1+ 1+ 1 \\ &= 9+4+ 4+ 2+ 2+ 2 \\ &=9+4+ 4+ 2+ 2+ 1+ 1 \\ &=9+4+ 4+ 2+ 1+ 1+ 1+ 1 \\ &=9+4+ 4+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=9+4+ 3+ 3+ 3+ 1 \\ &=9+4+ 3+ 3+ 2+ 2 \\ &=9+4+ 3+ 3+ 2+ 1+ 1 \\ &=9+4+ 3+ 3+ 1+ 1+ 1+ 1 \\ &=9+4+ 3+ 2+ 2+ 2+ 1 \\ &=9+4+ 3+ 2+ 2+ 1+ 1+ 1 \\ &=9+4+ 3+ 2+ 1+ 1+ 1+ 1+ 1 \\ &=9+4+ 3+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=9+4+ 2+ 2+ 2+ 2+ 2 \\ &=9+4+ 2+ 2+ 2+ 2+ 1+ 1 \\ &=9+4+ 2+ 2+ 2+ 1+ 1+ 1+ 1 \\ &=9+4+ 2+ 2+ 1+ 1+ 1+ 1+ 1+ 1 \\ &= 9+4+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=9+4+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=9+3+ 3+ 3+ 3+ 2 \\ &=9+3+ 3+ 3+ 3+ 1+ 1 \\ &= 9+3+ 3+ 3+ 2+ 2+ 1 \\ &=9+3+ 3+ 3+ 2+ 1+ 1+ 1 \\ &=9+3+ 3+ 3+ 1+ 1+ 1+ 1+ 1 \\ &=9+3+ 3+ 2+ 2+ 2+ 2 \\ &=9+3+ 3+ 2+ 2+ 2+ 1+ 1 \\ &=9+3+ 3+ 2+ 2+ 1+ 1+ 1+ 1 \\ &=9+3+ 3+ 2+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=9+3+ 3+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=9+3+ 2+ 2+ 2+ 2+ 2+ 1 \\ &=9+3+ 2+ 2+ 2+ 2+ 1+ 1+ 1 \\ &=9+3+ 2+ 2+ 2+ 1+ 1+ 1+ 1+ 1 \\ &=9+3+ 2+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=9+3+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=9+3+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=9+2+ 2+ 2+ 2+ 2+ 2+ 2 \\ &=9+2+ 2+ 2+ 2+ 2+ 2+ 1+ 1 \\ &= 9+2+ 2+ 2+ 2+ 2+ 1+ 1+ 1+ 1 \\ &=9+2+ 2+ 2+ 2+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=9+2+ 2+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=9+2+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=9+2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &= 9+1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=8+8+ 7 \\ &=8+8+ 6+ 1 \\ &=8+8+ 5+ 2 \\ &= 8+8+ 5+ 1+ 1 \\ &=8+8+ 4+ 3 \\ &=8+8+ 4+ 2+ 1 \\ &=8+8+ 4+ 1+ 1+ 1 \\ &=8+8+ 3+ 3+ 1 \\ &=8+8+ 3+ 2+ 2 \\ &=8+8+ 3+ 2+ 1+ 1 \\ &=8+8+ 3+ 1+ 1+ 1+ 1 \\ &= 8+8+ 2+ 2+2+ 1 \\ &=8+8+ 2+ 2+ 1+ 1+ 1 \\ &=8+8+ 2+ 1+ 1+ 1+ 1+ 1 \\ &=8+8+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &= 8+7+ 7+ 1 \\ &=8+7+ 6+ 2 \\ &=8+7+ 6+ 1+ 1 \\ &=8+7+ 5+ 3 \\ &=8+7+ 5+ 2+ 1 \\ &= 8+7+ 5+ 1+ 1+ 1 \\ &=8+7+ 4+ 4 \\ &=8+7+ 4+ 3+ 1 \\ &=8+7+ 4+ 2+ 2 \\ &=8+7+ 4+ 2+ 1+ 1 \\ &=8+7+ 4+ 1+ 1+ 1+ 1 \\ &= 8+7+ 3+ 3+ 2\\ &= 8+7+ 3+ 3+ 1+ 1 \\ &=8+7+ 3+ 2+ 2+ 1 \\ &=8+7+ 3+ 2+ 1+ 1+ 1 \\ &=8+7+ 3+ 1+ 1+ 1+ 1+ 1 \\ &=8+7+ 2+ 2+ 2+ 2 \\ &=8+7+ 2+ 2+ 2+ 1+ 1 \\ &=8+7+ 2+ 2+ 1+ 1+ 1+ 1 \\ &=8+7+ 2+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=8+7+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=8+6+ 6+ 3 \\ &=8+6+ 6+ 2+ 1 \\ &=8+6+ 6+ 1+ 1+ 1 \\ &=8+6+ 5+ 4 \\ &=8+6+ 5+ 3+ 1 \\ &=8+6+ 5+ 2+ 2 \\ &=8+6+ 5+ 2+ 1+ 1 \\ &=8+6+ 5+ 1+ 1+ 1+ 1 \\ &=8+6+ 4+ 4+ 1 \\ &=8+6+ 4+ 3+ 2 \\ &=8+6+ 4+ 3+ 1+ 1 \\ &= 8+6+ 4+ 2+ 2+ 1 \\ &=8+6+ 4+ 2+ 1+ 1+ 1 \\ &= 8+6+ 4+ 1+ 1+ 1+ 1+ 1 \\ &=8+6+ 3+ 3+ 3 \\ &=8+6+ 3+ 3+ 2+ 1 \\ &=8+6+ 3+ 3+ 1+ 1+ 1 \\ &=8+6+ 3+ 2+ 2+ 2 \\ &=8+6+ 3+ 2+ 2+ 1+ 1 \\ &=8+6+ 3 +2+ 1+ 1+ 1+ 1 \\ &=8+6+ 3+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=8+6+ 2+ 2+ 2+ 2+ 1 \\ &=8+6+ 2+ 2+ 2+ 1+ 1+ 1 \\ &=8+6+ 2+ 2+ 1+ 1+ 1+ 1+ 1 \\ &=8+6+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=8+6+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=8+5+ 5+ 5 \\ &=8+5+ 5+ 4+ 1 \\ &=8+5+ 5+ 3+ 2 \\ &=8+5+ 5+ 3+ 1+ 1 \\ &=8+5+ 5+ 2+ 2+ 1 \\ &=8+5+ 5+ 2+ 1+ 1+ 1 \\ &=8+5+ 5+ 1+ 1+ 1+ 1+ 1 \\ &=8+5+ 4+ 4+ 2 \\ &=8+5+ 4+ 4+ 1+ 1 \\ &=8+5+ 4+ 3+ 3 \\ &=8+5+ 4+ 3+ 2+ 1 \\ &=8+5+ 4+ 3+ 1+ 1+ 1 \\ &=8+5+ 4+ 2+ 2+ 2 \\ &=8+5+ 4+ 2+ 2+ 1+ 1 \\ &=8+5+ 4+ 2+ 1+ 1+ 1+ 1 \\ &=8+5+ 4+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=8+5+ 3+ 3+ 3+ 1 \\ &=8+5+ 3+ 3+ 2+ 2 \\ &= 8+5+ 3+ 3+ 2+ 1+ 1 \\ &=8+5+ 3+ 3+ 1+ 1+ 1+ 1 \\ &=8+5+ 3+ 2+ 2+ 2+ 1 \\ &=8+5+ 3+ 2+ 2+ 1+ 1+ 1 \\ &=8+5+ 3+ 2+ 1+ 1+ 1+ 1+ 1 \\ &=8+5+ 3+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=8+5+ 2+ 2+ 2+ 2+ 2 \\ &= 8+5+ 2+ 2+ 2+ 2+ 1+ 1 \\ &=8+5+ 2+ 2+ 2+ 1+ 1+ 1+ 1 \\ &=8+5+ 2+ 2+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=8+5+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=8+5+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=8+4+ 4+ 4+ 3 \\ &=8+4+ 4+ 4+ 2+ 1 \\ &=8+4 +4+ 4+ 1+ 1+ 1 \\ &=8+4+ 4+ 3+ 3+ 1 \\ &= 8+4+ 4+ 3+ 2+ 2 \\ &=8+4+ 4+ 3+ 2+ 1+ 1 \\ &=8+4+ 4+ 3+ 1+ 1+ 1+ 1 \\ &=8+4+ 4+ 2+ 2+ 2+ 1 \\ &=8+4+ 4+ 2+ 2+ 1+ 1+ 1 \\ &=8+4+ 4+ 2+ 1+ 1+ 1+ 1+ 1 \\ &= 8+4+ 4+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &= 8+4+ 3+ 3+ 3+ 2 \\ &=8+4+ 3+ 3+ 3+ 1+ 1 \\ &=8+4+ 3+ 3+ 2+ 2+ 1 \\ &=8+4+ 3+ 3+ 2+ 1+ 1+ 1 \\ &=8+4+ 3+ 3+ 1+ 1+ 1+ 1+ 1 \\ &=8+4+ 3+ 2+ 2+ 2+ 2 \\ &=8+4+ 3+ 2+ 2+ 2+ 1+ 1 \\ &=8+4+ 3+ 2+ 2+ 1+ 1+ 1+ 1 \\ &=8+4+ 3+ 2+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=8+4+ 3+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=8+4+ 2+ 2+ 2+ 2+ 2+ 1 \\ &=8+4+ 2+ 2+ 2+ 2+ 1+ 1+ 1 \\ &=8+4+ 2+ 2+ 2+ 1+ 1+ 1+ 1+ 1 \\ &=8+4+ 2+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=8+4+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=8+4+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=8+3+ 3+ 3+ 3+ 3 \\ &=8+3+ 3+ 3+ 3+ 2+ 1 \\ &=8+3+ 3+ 3+ 3+ 1+ 1+ 1 \\ &=8+3+ 3+ 3+ 2+ 2+ 2 \\ &= 8+3+ 3+ 3+ 2+ 2+ 1+ 1\\ &=8+3+ 3+ 3+ 2+ 1+ 1+ 1+ 1 \\ &=8+3+ 3+ 3+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=8+3+ 3+ 2+ 2+ 2+ 2+ 1 \\ &=8+3+ 3+ 2+ 2+ 2+ 1+ 1+ 1 \\ &=8+3+ 3+ 2+ 2+ 1+ 1+ 1+ 1+ 1 \\ &=8+3+ 3+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=8+3+ 3+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=8+3+ 2+ 2+ 2+ 2+ 2+ 2 \\ &=8+3+ 2+ 2+ 2+ 2+ 2+ 1+ 1 \\ &=8+3+ 2+ 2+ 2+ 2+ 1+ 1+ 1+ 1 \\ &=8+3+ 2+ 2+ 2+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=8+3+ 2+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=8+3+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=8+3+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=8+2+ 2+ 2+ 2+ 2+ 2+ 2+ 1 \\ &=8+2+ 2+ 2+ 2+ 2+ 2+ 1+ 1+ 1 \\ &=8+2+ 2+ 2+ 2+ 2+ 1+ 1+ 1+ 1+ 1 \\ &=8+2+ 2+ 2+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=8+2+ 2+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=8+2+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=8+2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=8+1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=7+7+ 7+ 2 \\ &= 7+7+ 7+ 1+ 1 \\ &=7+7+ 6+ 3 \\ &=7+7+ 6+ 2+ 1 \\ &=7+7+ 6+ 1+ 1+ 1 \\ &=7+7+ 5+ 4 \\ &=7+7+ 5+ 3+ 1 \\ &=7+7+ 5+ 2+ 2 \\ &=7+7+ 5+ 2+ 1+ 1 \\ &=7+7+ 5+ 1+ 1+ 1+ 1 \\ &=7+7+ 4+ 4+ 1 \\ &=7+7+ 4+ 3+ 2 \\ &=7+7+ 4+ 3+ 1+ 1 \\ &=7+7+ 4+ 2+ 2+ 1 \\ &=7+7+ 4+ 2+ 1+ 1+ 1 \\ &=7+7+ 4+ 1+ 1+ 1+ 1+ 1 \\ &=7+7+ 3+ 3+ 3 \\ &=7+7+ 3+ 3+ 2+ 1 \\ &=7+7+ 3+ 3+ 1+ 1+ 1 \\ &=7+7+ 3+ 2+ 2+ 2 \\ &=7+7+ 3+ 2+ 2+ 1+ 1 \\ &=7+7 +3+ 2+ 1+ 1+ 1+ 1 \\ &=7+7+ 3+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=7+7+ 2+ 2+ 2+ 2+ 1 \\ &=7+7+ 2+ 2+ 2+ 1+ 1+ 1 \\ &=7+7+ 2+ 2+ 1+ 1+ 1+ 1+ 1 \\ &=7+7+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=7+7+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=7+6+ 6+ 4 \\ &=7+6+ 6+ 3+ 1 \\ &=7+6+ 6+ 2+ 2 \\ &=7+6+ 6+ 2+ 1+ 1 \\ &= 7+6+ 6+ 1+ 1+ 1+ 1 \\ &=7+6+ 5+ 5 \\ &=7+6+ 5+ 4+ 1 \\ &=7+6+ 5+ 3+ 2 \\ &=7+6+ 5+ 3+ 1+ 1 \\ &=7+6+ 5+ 2+ 2+ 1 \\ &=7+6+ 5+ 2+ 1+ 1+ 1 \\ &=7+6+ 5+ 1+ 1+ 1+ 1+ 1 \\ &=7+6+ 4+ 4+ 2 \\ &=7+6+ 4+ 4+ 1+ 1 \\ &=7+6+ 4+ 3+ 3 \\ &=7+6+ 4+ 3+ 2+ 1 \\ &= 7+6+ 4+ 3+ 1+ 1+ 1 \\ &=7+6+ 4+ 2+ 2+ 2 \\ &=7+6+ 4+ 2+ 2+ 1+ 1 \\ &=7+6+ 4+ 2+ 1+ 1+ 1+ 1 \\ &=7+6+ 4+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=7+6+ 3+ 3+ 3+ 1 \\ &=7+6+ 3+ 3+ 2+ 2 \\ &=7+6+ 3+ 3+ 2+ 1+ 1 \\ &=7+6+ 3+ 3+ 1+ 1+ 1+ 1 \\ &=7+6+ 3+ 2+ 2+ 2+ 1 \\ &=7+6+ 3+ 2+ 2+ 1+ 1+ 1 \\ &=7+6+ 3+ 2+ 1+ 1+ 1+ 1+ 1 \\ &=7+6+ 3+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=7+6+ 2+ 2+ 2+ 2+ 2 \\ &=7+6+2+ 2+ 2+ 2+ 1+ 1 \\ &=7+6+ 2+ 2+ 2+ 1+ 1+ 1+ 1 \\ &=7+6+ 2+ 2+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=7+6+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=7+6+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=7+5+ 5+ 5+ 1 \\ &=7+5+ 5+ 4+ 2 \\ &=7+5+ 5+ 4+ 1+ 1 \\ &=7+5+ 5+ 3+ 3 \\ &=7+5+ 5+ 3+ 2+ 1 \\ &=7+5+ 5+ 3+ 1+ 1+ 1 \\ &= 7+5+ 5+ 2 +2+ 2 \\ &=7+5+ 5+ 2+ 2+ 1+ 1 \\ &=7+5+ 5+ 2+ 1+ 1+ 1+ 1 \\ &=7+5+ 5+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=7+5+ 4+ 4+ 3 \\ &=7+5+ 4+ 4+ 2+ 1 \\ &=7+5+ 4+ 4+ 1+ 1+ 1 \\ &=7+5+ 4+ 3+ 3+ 1 \\ &=7+5+ 4+ 3+ 2+ 2 \\ &=7+5+ 4+ 3+ 2+ 1+ 1 \\ &= 7+5+ 4+ 3+ 1+ 1+ 1+ 1 \\ &= 7+5+ 4+ 2+ 2+ 2+ 1 \\ &=7+5+ 4+ 2+ 2+ 1+ 1+ 1 \\ &=7+5+ 4+ 2+ 1+ 1+ 1+ 1+ 1 \\ &=7+5+ 4+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=7+5+ 3+ 3+ 3+ 2 \\ &=7+5+ 3+ 3+ 3+ 1+ 1 \\ &=7+5+ 3+ 3+ 2+ 2+ 1 \\ &=7+5+ 3+ 3+ 2+ 1+ 1+ 1 \\ &=7+5+ 3+ 3+ 1+ 1+ 1+ 1+ 1 \\ &=7+5+ 3+ 2+ 2+ 2+ 2 \\ &=7+5+ 3+ 2+ 2+ 2+ 1+ 1 \\ &=7+5+ 3+ 2+ 2+ 1+ 1+ 1+ 1 \\ &=7+5+ 3+ 2+ 1+ 1+ 1+ 1+ 1+ 1 \\ &= 7+5+ 3+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=7+5+ 2+ 2+ 2+ 2+ 2+ 1 \\ &=7+5+ 2+ 2+ 2+ 2+ 1+ 1+ 1 \\ &= 7+5+ 2+ 2+ 2+ 1+ 1+ 1+ 1+ 1 \\ &=7+5+ 2+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=7+5+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=7+5+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=7+4+4 +4 +4 \\ &= 7+4+ 4+ 4+ 3+ 1 \\ &=7+4+ 4+ 4+ 2+ 2 \\ &=7+4+ 4+ 4+ 2+ 1+ 1 \\ &=7+4+ 4+ 4+ 1+ 1+ 1+ 1 \\ &=7+4+ 4+ 3+ 3+ 2 \\ &=7+4+ 4+ 3+ 3+ 1+ 1 \\ &=7+4+ 4+ 3+ 2+ 2+ 1 \\ &= 7+4+ 4+ 3+ 2+ 1+ 1+ 1 \\ &=7+4+ 4+ 3+ 1+ 1+ 1+ 1+ 1 \\ &=7+4+ 4+ 2+ 2+ 2+ 2 \\ &=7+4+ 4+ 2+ 2+ 2+ 1+ 1 \\ &=7+4+ 4+ 2+ 2+ 1+ 1+ 1+ 1 \\ &=7+4+ 4+ 2+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=7+4+ 4+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=7+4+ 3+ 3+ 3+ 3 \\ &= 7+4+ 3+ 3+ 3+ 2+ 1 \\ &= 7+4+ 3+ 3+ 3+ 1+ 1+ 1 \\ &=7+4+ 3+ 3+ 2+ 2+ 2 \\ &=7+4+ 3+ 3+ 2+ 2+ 1+ 1 \\ &=7+4+ 3+ 3+ 2+ 1+ 1+ 1+ 1 \\ &=7+4+ 3+ 3+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=7+4+ 3+ 2+ 2+ 2+ 2+ 1 \\ &=7+4+ 3+ 2+ 2+ 2+ 1+ 1+ 1 \\ &=7+4+ 3+ 2+ 2+ 1+ 1+ 1+ 1+ 1 \\ &= 7+4+ 3+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=7+4+ 3+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=7+4+ 2+ 2+ 2+ 2+ 2+ 2 \\ &=7+4+ 2+ 2+ 2+ 2+ 2+ 1+ 1 \\ &= 7+4+ 2+ 2+ 2+ 2+ 1+ 1+ 1+ 1 \\ &=7+4+ 2+ 2+ 2+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=7+4+ 2+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=7+4+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=7+4+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=7+3+ 3+ 3+ 3+ 3+ 1 \\ &=7+3+ 3+ 3+ 3+ 2+ 2 \\ &=7+3+ 3+ 3+ 3+ 2+ 1+ 1 \\ &=7+3+ 3+ 3+ 3+ 1+ 1+ 1+ 1 \\ &=7+3+ 3+ 3+ 2+ 2+ 2+ 1 \\ &=7+3+ 3+ 3+ 2+ 2+ 1+ 1+ 1 \\ &=7+3+ 3+ 3+ 2+ 1+ 1+ 1+ 1+ 1 \\ &=7+3+ 3+ 3+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &= 7+3+ 3+ 2+ 2+ 2+ 2+ 2 \\ &=7+3+ 3+ 2+ 2+ 2+ 2+ 1+ 1 \\ &=7+3+ 3+ 2+ 2+ 2+ 1+ 1+ 1+ 1 \\ &=7+3+ 3+ 2+ 2+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=7+3+ 3+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=7+3+ 3+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=7+3+ 2+ 2+ 2+ 2+ 2+ 2+ 1 \\ &=7+3+ 2+ 2+ 2+ 2+ 2+ 1+ 1+ 1 \\ &= 7+3+ 2+ 2+ 2+ 2+ 1+ 1+ 1+ 1+ 1 \\ &=7+3+ 2+ 2+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=7+3+ 2+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=7+3+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=7+3+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=7+2+ 2+ 2+ 2+ 2+ 2+ 2+ 2 \\ &= 7+2+ 2+ 2+ 2+ 2+ 2+ 2+ 1+ 1 \\ &=7+2+ 2+ 2+ 2+ 2+ 2+ 1+ 1+ 1+ 1 \\ &=7+2+ 2+ 2+ 2+ 2+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=7+2+ 2+ 2+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=7+2+ 2+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=7+2+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=7+2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=7+1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=6+6+ 6+ 5 \\ &=6+6+ 6+ 4+ 1 \\ &=6+6+ 6+ 3+ 2 \\ &=6+6+ 6+ 3+ 1+ 1 \\ &=6+6+ 6+ 2+ 2+ 1 \\ &=6+6+ 6+ 2+ 1+ 1+ 1 \\ &=6+6+ 6+ 1+ 1+ 1+ 1+ 1 \\ &=6+6+ 5+ 5+ 1 \\ &=6+6+ 5+ 4+ 2 \\ &=6+6+ 5+ 4+ 1+ 1 \\ &=6+6+ 5+ 3+ 3 \\ &=6+6+ 5+ 3+ 2+ 1 \\ &=6+6+ 5+ 3+ 1+ 1+ 1 \\ &=6+6+ 5+ 2+ 2+ 2 \\ &=6+6+ 5+ 2+ 2+ 1+ 1 \\ &=6+6+ 5+ 2+ 1+ 1+ 1+ 1 \\ &=6+6+ 5+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=6+6+ 4+ 4+ 3 \\ &=6+6+ 4+ 4+ 2+ 1 \\ &=6+6+ 4+ 4+ 1+ 1+ 1 \\ &=6+6+ 4+ 3+ 3+ 1 \\ &=6+6+ 4+ 3+ 2+ 2 \\ &=6+6+ 4+ 3+ 2+ 1+ 1 \\ &=6+6+ 4+ 3+ 1+ 1+ 1+ 1 \\ &=6+6+ 4+ 2+ 2+ 2+ 1 \\ &=6+6+ 4+ 2+ 2+ 1+ 1+ 1 \\ &=6+6+ 4+ 2+ 1+ 1+ 1+ 1+ 1 \\ &=6+6+ 4+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=6+6+ 3+ 3+ 3+ 2 \\ &=6+6+ 3+ 3+ 3+ 1+ 1 \\ &=6+6+ 3+ 3+ 2+ 2+ 1 \\ &=6+6+ 3+ 3+ 2+ 1+ 1+ 1 \\ &=6+6+ 3+ 3+ 1+ 1+ 1+ 1+ 1 \\ &=6+6+ 3+ 2+ 2+ 2+ 2 \\ &=6+6+ 3+ 2+ 2+ 2+ 1+ 1 \\ &=6+6+ 3+ 2+ 2+ 1+ 1+ 1+ 1 \\ &=6+6+ 3+ 2+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=6+6+ 3+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=6+6+ 2+ 2+ 2+ 2+ 2+ 1 \\ &=6+6+ 2+ 2+ 2+ 2+ 1+ 1+ 1 \\ &=6+6+ 2+ 2+ 2+ 1+ 1+ 1+ 1+ 1 \\ &=6+6+ 2+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=6+6+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=6+6+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &= 6+5+ 5+ 5+ 2 \\ &=6+5+ 5+ 5+ 1+ 1 \\ &=6+5+ 5+ 4+ 3 \\ &=6+5+ 5+ 4+ 2+ 1 \\ &=6+5+ 5+ 4+ 1+ 1+ 1 \\ &=6+5+ 5+ 3+ 3+ 1 \\ &=6+5+ 5+ 3+ 2+ 2 \\ &=6+5+ 5+ 3+ 2+ 1+ 1 \\ &= 6+5+ 5+ 3+ 1+ 1+ 1+ 1 \\ &=6+5+ 5+ 2+ 2+ 2+ 1 \\ &=6+5+ 5+ 2+ 2+ 1+ 1+ 1 \\ &=6+5+ 5+ 2+ 1+ 1+ 1+ 1+ 1 \\ &=6+5+ 5+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=6+5+ 4+ 4+ 4 \\ &=6+5+ 4+ 4+ 3+ 1 \\ &=6+5+ 4+ 4+ 2+ 2 \\ &=6+5+ 4+ 4+ 2+ 1+ 1 \\ &=6+5+ 4+ 4+ 1+ 1+ 1+ 1 \\ &=6+5+ 4+ 3+ 3+ 2 \\ &= 6+5+ 4+ 3+ 3+ 1+ 1 \\ &=6+5+ 4+ 3+ 2+ 2+ 1 \\ &= 6+5+ 4+ 3+ 2+ 1+ 1+ 1 \\ &=6+5+ 4+ 3+ 1+ 1+ 1+ 1+ 1 \\ &=6+5+ 4+ 2+ 2+ 2+ 2 \\ &=6+5+ 4+ 2+ 2+ 2+ 1+ 1 \\ &=6+5+ 4+ 2+ 2+ 1+ 1+ 1+ 1 \\ &=6+5+ 4+ 2+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=6+5+ 4+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=6+5+ 3+ 3+ 3+ 3 \\ &=6+5+ 3+ 3+ 3+ 2+ 1 \\ &=6+5+ 3+ 3+ 3+ 1+ 1+ 1 \\ &=6+5+ 3+ 3+ 2+ 2+ 2 \\ &=6+5+ 3+ 3+ 2+ 2+ 1+ 1 \\ &=6+5+ 3+ 3+ 2+ 1+ 1+ 1+ 1 \\ &= 6+5+ 3+ 3+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=6+5+ 3+ 2+ 2+ 2+ 2+ 1 \\ &=6+5+ 3+ 2+ 2+ 2+ 1+ 1+ 1 \\ &=6+5+ 3+ 2+ 2+ 1+ 1+ 1+ 1+ 1 \\ &=6+5+ 3+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=6+5+ 3+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=6+5+ 2+ 2+ 2+ 2+ 2+ 2 \\ &=6+5+ 2+ 2+ 2+ 2+ 2+ 1+ 1 \\ &=6+5+ 2+ 2+ 2+ 2+ 1+ 1+ 1+ 1 \\ &=6+5+ 2+ 2+ 2+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=6+5+ 2+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &= 6+5+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &= 6+5+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &= 6+4+ 4+ 4+ 4+ 1 \\ &=6+4+ 4+ 4+ 3+ 2 \\ &=6+4+ 4+ 4+ 3+ 1+ 1 \\ &=6+4+ 4+ 4+ 2+ 2+ 1 \\ &=6+4+ 4+ 4+ 2+ 1+ 1+ 1 \\ &=6+4 +4 +4 +1+ 1+ 1+ 1+ 1 \\ &=6+4+ 4+ 3+ 3+ 3 \\ &=6+4+ 4+ 3+ 3+ 2+ 1 \\ &=6+4+ 4+ 3+ 3+ 1+ 1+ 1 \\ &= 6+4+ 4+ 3+ 2+ 2+ 2 \\ &=6+4+ 4+ 3+ 2+ 2+ 1+ 1 \\ &=6+4+ 4+ 3+ 2+ 1+ 1+ 1+ 1 \\ &=6+4+ 4+ 3+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=6+4+ 4+ 2+ 2+ 2+ 2+ 1 \\ &=6+4+ 4+ 2+ 2+ 2+ 1+ 1+ 1 \\ &= 6+4+ 4+ 2+ 2+ 1+ 1+ 1+ 1+ 1 \\ &=6+4+ 4+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=6+4+ 4+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=6+4+ 3+ 3+ 3+ 3+ 1 \\ &=6+4+ 3+ 3+ 3+ 2+ 2 \\ &= 6+4+ 3+ 3+ 3+ 2+ 1+ 1 \\ &=6+4+ 3+ 3+ 3+ 1+ 1+ 1+ 1 \\ &= 6+4+ 3+ 3+ 2+ 2+ 2+ 1 \\ &=6+4+ 3+ 3+ 2+ 2+ 1+ 1+ 1 \\ &=6+4+ 3+ 3+ 2+ 1+ 1+ 1+ 1+ 1 \\ &=6+4+ 3+ 3+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &= 6+4+ 3+ 2+ 2+ 2+ 2+ 2 \\ &=6+4+ 3+ 2+ 2+ 2+ 2+ 1+ 1 \\ &=6+4+ 3+ 2+ 2+ 2+ 1+ 1+ 1+ 1 \\ &=6+4+ 3+ 2+ 2+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=6+4+ 3+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=6+4+ 3+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=6+4+ 2+ 2+ 2+ 2+ 2+ 2+ 1 \\ &=6+4+ 2+ 2+ 2+ 2+ 2+ 1+ 1+ 1 \\ &=6+4+ 2+ 2+ 2+ 2+ 1+ 1+ 1+ 1+ 1 \\ &=6+4+ 2+ 2+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=6+4+ 2+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=6+4+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=6+4+ 1+ 1+ 1+ 1+ 1+ 1+ 1 +1+ 1+ 1+ 1+ 1+ 1 \\ &=6+3+ 3+ 3+ 3+ 3+ 2 \\ &=6+3+ 3+ 3+ 3+ 3+ 1+ 1 \\ &=6+3+ 3+ 3+ 3+ 2+ 2+ 1 \\ &= 6+3+ 3+ 3+ 3+ 2+ 1+ 1+ 1 \\ &=6+3+ 3+ 3+ 3+ 1+ 1+ 1+ 1+ 1 \\ &=6+3+ 3+ 3+ 2+ 2+ 2+ 2 \\ &= 6+3+ 3+ 3+ 2+ 2+ 2+ 1+ 1 \\ &=6+3+ 3+ 3+ 2+ 2+ 1+ 1+ 1+ 1 \\ &=6+3+ 3+ 3+ 2+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=6+3+ 3+ 3+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &= 6+3+ 3+ 2+ 2+ 2+ 2+ 2+ 1 \\ &=6+3+ 3+ 2+ 2+ 2+ 2+ 1+ 1+ 1 \\ &=6+3+ 3+ 2+ 2+ 2+ 1+ 1+ 1+ 1+ 1 \\ &=6+3+ 3+ 2+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=6+3+ 3+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=6+3+ 3+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=6+3+ 2+ 2+ 2+ 2+ 2+ 2+ 2 \\ &=6+3+ 2+ 2+ 2+ 2 +2+ 2+ 1+ 1 \\ &=6+3+ 2+ 2+ 2+ 2+ 2+ 1+ 1+ 1+ 1 \\ &= 6+3+ 2+ 2+ 2+ 2+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=6+3+ 2+ 2+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=6+3+ 2+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=6+3+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &= 6+3+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=6+2+ 2+ 2+ 2+ 2+ 2+ 2+ 2+ 1 \\ &=6+2+ 2+ 2+ 2+ 2+ 2+ 2+ 1+ 1+ 1 \\ &=6+2+ 2+ 2+ 2+ 2+ 2+ 1+ 1+ 1+ 1+ 1 \\ &=6+2+ 2+ 2+ 2+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=6+2+ 2+ 2+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=6+2+ 2+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=6+2+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &= 6+2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=6+1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=5+5+ 5+ 5+ 3 \\ &=5+5+ 5+ 5+ 2+ 1 \\ &=5+5+ 5+ 5+ 1+ 1+ 1 \\ &=5+5+ 5+ 4+ 4 \\ &=5+5+ 5+ 4+ 3+ 1 \\ &=5+5+ 5+ 4+ 2+ 2 \\ &=5+5+ 5+ 4+ 2+ 1+ 1 \\ &=5+5+ 5+ 4+ 1+ 1+ 1+ 1 \\ &=5+5+ 5+ 3+ 3+ 2 \\ &=5+5+ 5+ 3+ 3+ 1+ 1 \\ &=5+5+ 5+ 3+ 2+ 2+ 1 \\ &=5+5+ 5+ 3+ 2+ 1+ 1+ 1 \\ &=5+5+ 5+ 3+ 1+ 1+ 1+ 1+ 1 \\ &=5+5+ 5+ 2+ 2+ 2+ 2 \\ &=5+5+ 5+ 2+ 2+ 2+ 1+ 1 \\ &=5+5+ 5+ 2+ 2+ 1+ 1+ 1+ 1 \\ &=5+5+ 5+ 2+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=5+5+ 5+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=5+5+ 4+ 4+ 4+ 1 \\ &= 5+5+ 4+ 4+ 3+ 2 \\ &=5+5+ 4+ 4+ 3+ 1+ 1 \\ &=5+5+ 4+ 4+ 2+ 2+ 1 \\ &=5+5+ 4+ 4+ 2+ 1+ 1+ 1 \\ &=5+5+ 4+ 4+ 1+ 1+ 1+ 1+ 1 \\ &=5+5+ 4+ 3+ 3+ 3 \\ &=5+5+ 4+ 3+ 3+ 2+ 1 \\ &=5+5+ 4+ 3+ 3+ 1+ 1+ 1 \\ &=5+5+ 4+ 3+ 2+ 2+ 2 \\ &=5+5+ 4+ 3+ 2+ 2+ 1+ 1 \\ &=5+5+ 4+ 3+ 2+ 1+ 1+ 1+ 1 \\ &=5+5+ 4+ 3+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=5+5+ 4+ 2+ 2+ 2+ 2+ 1 \\ &=5+5+ 4+ 2+ 2+ 2+ 1+ 1+ 1 \\ &=5+5+ 4+ 2+ 2+ 1+ 1+ 1+ 1+ 1 \\ &=5+5+ 4+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=5+5+ 4+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=5+5+ 3+ 3+ 3+ 3+ 1 \\ &=5+5+ 3+ 3+ 3+ 2+ 2 \\ &=5+5+ 3+ 3+ 3+ 2+ 1+ 1 \\ &=5+5+ 3+ 3+ 3+ 1+ 1+ 1+ 1 \\ &=5+5+ 3+ 3+ 2+ 2+ 2+ 1 \\ &=5+5+ 3+ 3+ 2+ 2+ 1+ 1+ 1 \\ &=5+5+ 3+ 3+ 2+ 1+ 1+ 1+ 1+ 1 \\ &=5+5+ 3+ 3+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=5+5+ 3+ 2+ 2+ 2+ 2+ 2 \\ &=5+5+ 3+ 2+ 2+ 2+ 2+ 1+ 1 \\ &=5+5+ 3+ 2+ 2+ 2+ 1+ 1+ 1+ 1 \\ &=5+5+ 3+ 2+ 2+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=5+5+ 3+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &= 5+5+ 3+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=5+5+ 2+ 2+ 2+ 2+ 2+ 2+ 1 \\ &=5+5+ 2+ 2+ 2+ 2+ 2+ 1+ 1+ 1 \\ &=5+5+ 2+ 2+ 2+ 2+ 1+ 1+ 1+ 1+ 1 \\ &=5+5+ 2+ 2+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=5+5+ 2+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=5+5+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=5+5+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &= 5+4+ 4+ 4+ 4+ 2 \\ &=5+4+ 4+ 4+ 4+ 1+ 1 \\ &=5+4+ 4+ 4+ 3+ 3 \\ &=5+4+ 4+ 4+ 3+ 2+ 1 \\ &=5+4+ 4+ 4+ 3+ 1+ 1+ 1 \\ &=5+4+ 4+ 4+ 2+ 2+ 2 \\ &=5+4+ 4+ 4+ 2+ 2+ 1+ 1 \\ &= 5+4+ 4+ 4+ 2+ 1+ 1+ 1+ 1 \\ &=5+4+ 4+ 4+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=5+4+ 4+ 3+ 3+ 3+ 1 \\ &=5+4+ 4+ 3+ 3+ 2+ 2 \\ &=5+4+ 4+ 3+ 3+ 2+ 1+ 1 \\ &=5+4+ 4+ 3+ 3+ 1+ 1+ 1+ 1 \\ &=5+4+ 4+ 3+ 2+ 2+ 2+ 1 \\ &= 5+4+ 4+ 3+ 2+ 2+ 1+ 1+ 1 \\ &=5+4+ 4+ 3+ 2+ 1+ 1+ 1+ 1+ 1 \\ &=5+4+ 4+ 3+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=5+4+ 4+ 2+ 2+ 2+ 2+ 2 \\ &=5+4+ 4+ 2+ 2+ 2+ 2+ 1+ 1 \\ &=5+4+ 4+ 2+ 2+ 2+ 1+ 1+ 1+ 1 \\ &=5+4+ 4+ 2+ 2+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=5+4+ 4+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=5+4+ 4+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=5+4+ 3+ 3+ 3+ 3+ 2 \\ &=5+4+ 3+ 3+ 3+ 3+ 1+ 1 \\ &=5+4+ 3+ 3+ 3+ 2+ 2+ 1 \\ &=5+4+ 3+ 3+ 3+ 2+ 1+ 1+ 1 \\ &= 5+4+ 3+ 3+ 3+ 1+ 1+ 1+ 1+ 1 \\ &=5+4+ 3+ 3+ 2+ 2+ 2+ 2 \\ &=5+4+ 3+ 3+ 2+ 2+ 2+ 1+ 1 \\ &=5+4+ 3+ 3+ 2+ 2+ 1+ 1+ 1+ 1 \\ &=5+4+ 3+ 3+ 2+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=5+4+ 3+ 3+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=5+4+ 3+ 2+ 2+ 2+ 2+ 2+ 1 \\ &=5+4+ 3+ 2+ 2+ 2+ 2+ 1+ 1+ 1 \\ &=5+4+ 3+ 2+ 2+ 2+ 1+ 1+ 1+ 1+ 1 \\ &=5+4+ 3+ 2+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &= 5+4+ 3+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &= 5+4+ 3+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=5+4+ 2+ 2+ 2+ 2+ 2+ 2+ 2 \\ &=5+4+ 2+ 2+ 2+ 2+ 2+ 2+ 1+ 1 \\ &=5+4+ 2+ 2+ 2+ 2+ 2+ 1+ 1+ 1+ 1 \\ &=5+4+ 2+ 2+ 2+ 2+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=5+4+ 2+ 2+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=5+4+ 2+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=5+4+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=5+4+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=5+3+ 3+ 3+ 3+ 3+ 3 \\ &=5+3+ 3+ 3+ 3+ 3+ 2+ 1 \\ &=5+3+ 3+ 3+ 3+ 3+ 1+ 1+ 1 \\ &=5+3+ 3+ 3+ 3+ 2+ 2+ 2 \\ &=5+3+ 3+ 3+ 3+ 2+ 2+ 1+ 1 \\ &=5+3+ 3+ 3+ 3+ 2+ 1+ 1+ 1+ 1 \\ &=5+3+ 3+ 3+ 3+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=5+3+ 3+ 3+ 2+ 2+ 2+ 2+ 1 \\ &=5+3+ 3+ 3+ 2+ 2+ 2+ 1+ 1+ 1 \\ &=5+3+ 3+ 3+ 2+ 2+ 1+ 1+ 1+ 1+ 1 \\ &=5+3+ 3+ 3+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=5+3+ 3+ 3+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=5+3+ 3+ 2+ 2+ 2+ 2+ 2+ 2 \\ &=5+3+ 3+ 2+ 2+ 2+ 2+ 2+ 1+ 1 \\ &=5+3+ 3+ 2+ 2+ 2+ 2+ 1+ 1+ 1+ 1 \\ &=5+3+ 3+ 2+ 2+ 2+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=5+3+ 3+ 2+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &= 5+3+ 3+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=5+3+ 3+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=5+3+ 2+ 2+ 2+ 2+ 2+ 2+ 2+ 1 \\ &=5+3+ 2+ 2+ 2+ 2+ 2+ 2+ 1+ 1+ 1 \\ &=5+3+ 2+ 2+ 2+ 2+ 2+ 1+ 1+ 1+ 1+ 1 \\ &= 5+3+ 2+ 2+ 2+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=5+3+ 2+ 2+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &= 5+3+ 2+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=5+3+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=5+3+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=5+2+ 2+ 2+ 2+ 2+ 2+ 2+ 2+ 2 \\ &=5+2+ 2+ 2+ 2+ 2+ 2+ 2+ 2+ 1+ 1 \\ &=5+2+ 2+ 2+ 2+ 2+ 2+ 2+ 1+ 1+ 1+ 1 \\ &=5+2+ 2+ 2+ 2+ 2+ 2+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=5+2+ 2+ 2+ 2+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=5+2+ 2+ 2+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=5+2+ 2+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=5+2+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=5+2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=5+1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=4+4+ 4+ 4+ 4+ 3 \\ &=4+4+ 4+ 4+ 4+ 2+ 1 \\ &=4+4+ 4+ 4+ 4+ 1+ 1+ 1 \\ &=4+4+ 4+ 4+ 3+ 3+ 1 \\ &=4+4+ 4+ 4+ 3+ 2+ 2 \\ &=4+4+ 4+ 4+ 3+ 2+ 1+ 1 \\ &=4+4+ 4+ 4+ 3+ 1+ 1+ 1+ 1 \\ &=4+4+ 4+ 4+ 2+ 2+ 2+ 1 \\ &=4+4+ 4+ 4+ 2+ 2+ 1+ 1+ 1 \\ &=4+4+ 4+ 4+ 2+ 1+ 1+ 1+ 1+ 1 \\ &=4+4+ 4+ 4+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=4+4+ 4+ 3+ 3+ 3+ 2 \\ &=4+4+ 4+ 3+ 3+ 3+ 1+ 1 \\ &=4+4+ 4+ 3+ 3+ 2+ 2+ 1 \\ &=4+4+ 4+ 3+ 3+ 2+ 1+ 1+ 1 \\ &=4+4+ 4+ 3+ 3+ 1+ 1+ 1+ 1+ 1 \\ &=4+4+ 4+ 3+ 2+ 2+ 2+ 2 \\ &=4+4+ 4+ 3+ 2+ 2+ 2+ 1+ 1 \\ &=4+4+ 4+ 3+ 2+ 2+ 1+ 1+ 1+ 1 \\ &=4+4+ 4+ 3+ 2+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=4+4+ 4+ 3+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &= 4+4+ 4+ 2+ 2+ 2+ 2+ 2+ 1 \\ &=4+4+ 4+ 2+ 2+ 2+ 2+ 1+ 1+ 1 \\ &=4+4+ 4+ 2+ 2+ 2+ 1+ 1+ 1+ 1+ 1 \\ &=4+4+ 4+ 2+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=4+4+ 4+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=4+4+ 4+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=4+4+ 3+ 3+ 3+ 3+ 3 \\ &=4+4+ 3+ 3+ 3+ 3+ 2+ 1 \\ &=4+4+ 3+ 3+ 3+ 3+ 1+ 1+ 1 \\ &=4+4+ 3+ 3+ 3+ 2+ 2+ 2 \\ &=4+4+ 3+ 3+ 3+ 2+ 2+ 1+ 1 \\ &=4+4+ 3+ 3+ 3+ 2+ 1+ 1+ 1+ 1 \\ &=4+4+ 3+ 3+ 3+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=4+4+ 3+ 3+ 2+ 2+ 2+ 2+ 1 \\ &=4+4+ 3+ 3+ 2+ 2+ 2+ 1+ 1+ 1 \\ &=4+4+ 3+ 3+ 2+ 2+ 1+ 1+ 1+ 1+ 1 \\ &=4+4+ 3+ 3+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=4+4+ 3+ 3+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=4+4+ 3+ 2+ 2+ 2+ 2+ 2+ 2 \\ &=4+4+ 3+ 2+ 2+ 2+ 2+ 2+ 1+ 1 \\ &=4+4+ 3+ 2+ 2+ 2+ 2+ 1+ 1+ 1+ 1 \\ &=4+4+ 3+ 2+ 2+ 2+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=4+4+ 3+ 2+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=4+4+ 3+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=4+4+ 3+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=4+4+ 2+ 2+ 2+ 2+ 2+ 2+ 2+ 1 \\ &=4+4+ 2+ 2+ 2+ 2+ 2+ 2+ 1+ 1+ 1 \\ &=4+4+ 2+ 2+ 2+ 2+ 2+ 1+ 1+ 1+ 1+ 1 \\ &=4+4+ 2+ 2+ 2+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=4+4+ 2+ 2+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=4+4+ 2+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=4+4+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=4+4+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=4+3+ 3+ 3+ 3+ 3+ 3+ 1 \\ &=4+3+ 3+ 3+ 3+ 3+ 2+ 2 \\ &=4+3+ 3+ 3+ 3+ 3+ 2+ 1+ 1 \\ &=4+3+ 3+ 3+ 3+ 3+ 1+ 1+ 1+ 1 \\ &=4+3+ 3+ 3+ 3+ 2+ 2+ 2+ 1 \\ &=4+3+ 3+ 3+ 3+ 2+ 2+ 1+ 1+ 1 \\ &=4+3+ 3+ 3+ 3+ 2+ 1+ 1+ 1+ 1+ 1 \\ &=4+3+ 3+ 3+ 3+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &= 4+3+ 3+ 3+ 2+ 2+ 2+ 2+ 2 \\ &=4+3+ 3+ 3+ 2+ 2+ 2+ 2+ 1+ 1 \\ &=4+3+ 3+ 3+ 2+ 2+ 2+ 1+ 1+ 1+ 1 \\ &=4+3+ 3+ 3+ 2+ 2+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=4+3+ 3+ 3+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=4+3+ 3+ 3+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &= 4+3+ 3+ 2+ 2+ 2+ 2+ 2+ 2+ 1 \\ &= 4+3+ 3+ 2+ 2+ 2+ 2+ 2+ 1+ 1+ 1 \\ &=4+3+ 3+ 2+ 2+ 2+ 2+ 1+ 1+ 1+ 1+ 1 \\ &=4+3+ 3+ 2+ 2+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=4+3+ 3+ 2+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=4+3+ 3+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=4+3+ 3+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=4+3+ 2+ 2+ 2+ 2+ 2+ 2+ 2+ 2 \\ &=4+3+2+2+2+2+2+2+2+1+1\end{align}

これ以上、１つの数式では読み込まないようなので仕切り直し。

\begin{align}23&=4+3+ 2+ 2+ 2+ 2+ 2+ 2 +1+ 1+ 1+ 1\\ &=4+3+ 2+ 2+ 2+ 2+ 2+ 1+ 1+ 1+ 1+ 1+ 1\\ &=4+3+ 2+ 2+ 2+ 2+ 1+ 1+ 1 +1+ 1+ 1+ 1+ 1 \\ &=4+3+ 2+ 2+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=4+3+ 2+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &= 4+3+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &= 4+3+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=4+2+ 2+ 2+ 2+ 2+ 2+ 2+ 2+ 2+ 1 \\ &=4+2+ 2+ 2+ 2+ 2+ 2+ 2+ 2+ 1+ 1+ 1 \\ &=4+2+ 2 +2 +2 +2 +2 +2 +1+ 1+ 1+ 1+ 1 \\ &=4+2+ 2 +2 +2+ 2+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=4+2+ 2+ 2+ 2+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=4+2+ 2+ 2+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=4+2+ 2+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ +1 +1 +1 +1 +1 +1 \\ &=4+2+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=4+2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 +1+ 1+ 1+ 1 +1 \\ &=4+1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 +1 +1 +1 +1 \\ &= 3+3+ 3+ 3+ 3+ 3+ 3+ 2 \\ &= 3+3+ 3+ 3+ 3+ 3+ 3+ 1+ 1\\ &=3+3+ 3+ 3+ 3+ 3+ 2+ 2+ 1 \\ &=3+3+ 3+ 3+ 3+ 3+ 2+ 1+ 1+ 1 \\ &=3+3+ 3+ 3+ 3+ 3+ 1+ 1+ 1+ 1+ 1 \\ &= 3+3+ 3+ 3+ 3+ 2+ 2+ 2+ 2 \\ &=3+3+ 3+ 3+ 3+ 2+ 2+ 2+ 1+ 1 \\ &=3+3+ 3+ 3+ 3+ 2+ 2+ 1+ 1+ 1+ 1 \\ &= 3+3+ 3+ 3+ 3+ 2+ 1+ 1+ 1+ 1+ 1+ 1 \\ &= 3+3+ 3+ 3+ 3+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=3+3+ 3+ 3 +2+ 2+ 2+ 2+ 2+ 1 \\ &=3+3+ 3+ 3+ 2+ 2+ 2+ 2+ 1+ 1+ 1 \\ &=3+3+ 3+ 3+ 2+ 2+ 2+ 1+ 1+ 1+ 1+ 1 \\ &=3+3+ 3+ 3+ 2+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &= 3+3+ 3+ 3+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=3+3+ 3+ 3+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=3+3+ 3+ 2+ 2+ 2+ 2+ 2+ 2+ 2 \\ &=3+3+ 3+ 2+ 2+ 2+ 2+ 2+ 2+ 1+ 1 \\ &=3+3+ 3+ 2+ 2+ 2+ 2+ 2+ 1+ 1+ 1+ 1 \\ &= 3+3+ 3+ 2+ 2+ 2+ 2+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=3+3+ 3+ 2+ 2+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=3+3+ 3+ 2+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=3+3+ 3+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=3+3+ 3+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=3+3+ 2+ 2+ 2+ 2+ 2+ 2+ 2+ 2+ 1 \\ &=3+3+ 2+ 2+ 2+ 2+ 2+ 2+ 2+ 1+ 1+ 1\\ &=3+3+ 2+ 2+ 2+ 2+ 2+ 2+ 1+ 1+ 1+ 1+ 1 \\ &=3+3+ 2+ 2+ 2+ 2+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=3+3+ 2+ 2+ 2+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=3+3+ 2+ 2+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=3+3+ 2+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=3+3+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=3+3+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1\\ &=3+2+ 2+ 2+ 2+ 2+ 2+ 2+ 2+ 2+ 2 \\ &=3+2+ 2+ 2+ 2+ 2+ 2+ 2+ 2+ 2+ 1+ 1 \\ &=3+2+ 2+ 2+ 2+ 2+ 2+ 2+ 2+ 1+ 1+ 1+ 1 \\ &=3+2+ 2+ 2+ 2+ 2+ 2+ 2+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=3+2+ 2+ 2+ 2+ 2+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=3+2 +2 +2+ 2+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=3+2 +2 +2 +2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=3+2+ 2+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=3+2+ 2+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=3+2+1 +1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=3+1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1 \\ &=2+2+2+2+2+2+2+2+2+2+2+1\\ &=2+2+2+2+2+2+2+2+2+2+1+1+1\\ &=2+2+2+2+2+2+2+2+2+1+1+1+1+1\\ &=2+2+2+2+2+2+2+2+1+1+1+1+1+1+1\\ &=2+2+2+2+2+2+2+1+1+1+1+1+1+1+1+1\\ &=2+2+2+2+2+2+1+1+1+1+1+1+1+1+1+1+1\\ &=2+2+2+2+2+1+1+1+1+1+1+1+1+1+1+1+1+1\\ &=2+2+2+2+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1\\ &=2+2+2+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1\\ &=2+2+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1\\ &=2+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1\\ &=1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1 \end{align}

と書き下したときに現れる+の個数が8951ですね。