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Best Known Lower Bound to the Maximum
Problem Set #1
n/d 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%
20

4,135

3,104

7,880

5,750

8,925

7,457

9,665

6,394

9,915

12,982
30 4,547 7,132 13,735 11,163 15,134 13,260 16,103 20,770 19,270 21,141
40 8,753 17,334 17,319 23,558 27,005 22,140 26,249 30,775 38,519 39,014
50 12,037 19,161 32,079 32,929 40,101 33,969 39,198 41,483 45,641 53,462
60 22,333 27,828 36,547 45,747 37,740 42,266 57,386 55,636 56,640 65,808
70 26,276 25,438 40,669 44,012 50,521 62,571 55,667 70,494 81,555 81,186
80 26,647 39,337 56,301 55,133 86,562 80,746 85,984 96,508 85,055 84,109
90 39,631 55,622 70,663 73,690 83,672 118,152 93,554 101,924 101,581 103,821
100 36,544 68,109 71,697 93,205 91,954 102,816 117,522 133,363 131,785 159,436
Problem Set #2
n/d 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%
20

3,447

7,682

6,253

8,144

4,848

8,280

11,395

11,162

9,013

14,151
30 4,094 9,643 11,583 11,217 19,079 18,907 16,145 20,740 20,077 16,290
40 10,664 12,562 17,022 21,362 14,916 29,214 27,535 34,771 29,689 31,053
50 13,574 19,208 29,522 36,384 26,311 35,694 39,567 52,949 44,255 47,421
60 19,719 25,760 34,056 41,411 40,327 46,020 54,506 62,806 43,447 54,772
70 20,822 37,624 44,140 51,322 59,392 53,220 78,847 72,831 79,521 79,012
80 30,039 57,029 56,046 60,594 63,193 78,579 70,784 90,182 91,869 87,425
90 39,578 54,260 63,851 79,137 99,326 86,962 118,065 106,213 107,961 107,966
100 37,208 56,754 62,100 81,672 107,514 88,321 141,470 125,854 122,122 120,499
Problem Set #3
n/d 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%
20

2,269

6,527

3,680

8,476

12,224

6,916

10,879

9,924

10,161

14,503
30 8,212 8,795 11,758 17,845 16,146 18,055 17,048 16,609 20,922 18,395
40 10,807 13,928 17,998 22,502 20,637 19,409 29,317 28,135 31,676 40,975
50 14,492 15,670 27,687 32,461 37,266 35,245 42,955 47,070 45,290 51,773
60 14,175 29,267 29,088 38,200 33,726 43,673 53,473 57,908 61,728 58,923
70 24,095 39,800 41,222 54,432 62,343 72,307 50,919 77,168 76,358 90,510
80 26,210 41,758 47,712 48,460 83,736 79,703 77,200 76,395 83,783 104,436
90 36,806 49,817 60,973 86,050 82,002 101,449 104,899 95,127 85,028 92,828
100 47,179 49,134 63,898 97,566 101,945 106,737 94,377 140,934 132,031 161,291
Problem Set #4
n/d 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%
20

3,054

2,541

4,376

8,836

6,159

7,780

14,651

14,791

9,897

8,170
30 6,897 9,814 8,832 16,369 13,583 16,166 10,441 19,912 18,241 16,065
40 8,549 11,909 21,122 19,297 25,504 24,155 34,152 31,230 23,597 34,324
50 10,868 26,272 23,686 20,687 29,276 38,968 41,342 44,621 38,545 52,864
60 18,340 21,962 33,151 46,650 50,304 42,621 48,196 54,195 51,463 74,196
70 17,766 41,024 40,130 48,559 77,360 45,534 67,964 75,024 73,116 72,111
80 33,395 35,156 68,085 64,961 70,729 84,992 75,514 95,177 103,574 81,198
90 35,931 44,139 63,390 87,955 82,026 92,639 75,073 101,678 96,128 122,162
100 37,153 67,265 75,665 74,947 102,404 81,268 100,852 109,566 116,459 172,487
Problem Set #5
n/d 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%
20

4,342

6,710

6,466

8,550

10,493

11,942

16,178

12,692

13,515

15,315
30 3,660 7,668 10,594 12,061 13,432 12,704 16,601 18,417 18,660 24,246
40 10,556 15,436 18,568 16,412 22,504 29,310 33,097 41,500 30,521 31,545
50 15,498 16,280 25,487 33,587 20,562 34,157 41,457 36,485 39,699 49,119
60 25,055 21,742 48,643 30,540 57,820 52,084 67,020 64,833 68,222 49,145
70 24,854 40,151 46,064 53,359 53,029 61,782 77,860 63,951 82,207 61,011
80 28,205 47,365 50,883 67,776 75,189 84,182 75,155 109,608 96,597 89,154
90 38,899 65,446 66,180 79,250 74,961 88,767 96,259 109,633 107,325 121,997
100 38,744 72,007 68,001 108,514 83,877 104,716 117,737 118,397 126,867 113,410
Problem Set #6
n/d 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%
20

4,444

4,124

5,353

6,237

5,764

10,431

13,748

11,332

10,934

11,253
30 4,365 8,010 10,695 12,458 15,072 23,057 20,279 14,278 19,624 29,018
40 10,801 12,907 16,858 22,887 22,731 20,816 29,976 29,159 31,002 32,710
50 15,629 20,784 24,653 35,962 41,880 34,475 41,883 38,921 38,054 35,248
60 20,112 32,319 31,587 39,133 39,967 47,664 54,158 62,578 59,974 67,994
70 24,198 31,609 38,771