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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 48,011 47,383 75,836 71,592 72,304 99,157 74,569
80 31,961 42,520 60,586 49,704 81,427 79,224 104,533 85,833 101,428 99,655
90 43,817 59,711 55,978 66,127 90,709 82,860 95,798 110,329 81,200 122,828
100 47,667 62,907 80,193 75,962 110,432 96,699 106,064 120,609 132,136 130,630
Problem Set #7
n/d 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%
20

2,973

4,098

7,137

7,810

4,632

11,663

8,383

9,878

10,194

13,147
30 5,498 5,371 11,686 16,178 15,578 17,867 21,495 17,681 25,107 19,843
40 13,958 18,376 17,283 21,487 23,383 34,113 31,857 29,651 31,049 51,444
50 16,915 19,930 22,173 34,473 41,804 39,331 39,300 62,659 48,221 45,578
60 16,445 24,156 33,517 39,769 47,096 34,893 58,591 50,664 61,411 67,264
70 25,770 29,838 53,884 49,648 50,102 90,474 50,123 64,552 98,189 91,524
80 24,520 48,733 48,253 52,584 71,364 85,069 87,055 77,789 95,770 91,359
90 30,797 53,165 50,641 72,733 121,579 91,907 89,390 90,240 94,048 97,537
100 45,969 66,521 76,005 87,168 104,233 128,824 113,642 117,057 132,983 136,959
Problem Set #8
n/d 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%
20

2,434

3,976

4,358

7,386

7,018

7,527

8,243

10,375

11,364

9,406
30 9,291 7,904 12,028 12,543 20,363 16,180 16,519 20,014 16,718 20,869
40 6,117 13,336 15,699 17,165 18,547 26,350 29,315 20,957 24,648 31,657
50 18,035 29,380 21,760 23,174 35,954 35,464 39,895 37,922 40,862 35,001
60 15,704 26,602 34,598 41,441 42,968 41,363 70,337 50,157 58,253 67,723
70 19,606 32,639 44,907 42,570 52,535 64,763 72,469 78,403 83,389 74,475
80 26,835 40,651 54,221 67,749 70,953 80,640 82,631 96,991 103,377 109,223
90 41,457 48,730 62,371 85,462 81,022 91,633 114,826 100,423 107,495 164,480
100 42,214 59,601 67,486 90,406 84,407 114,396 145,882 126,349 138,858 156,735
Problem Set #9
n/d 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%
20

3,188

7,301

5,183

8,248

8,800

11,985

11,098

8,583

10,100

9,734
30 3,581 11,664 12,384 10,801 11,159 19,854 13,613 21,654 16,514 29,932
40 11,991 20,354 17,178 22,676 22,835 22,955 24,114 30,876 35,490 35,168
50 15,241 15,495 17,370 34,850 30,758 34,422 37,056 49,929 52,823 47,551
60 11,627 25,184 42,446 38,798 44,748 47,255 49,709 58,299 61,263 67,098
70 24,433 38,050 48,949 49,098 58,413 58,291 63,586 92,337 73,250 66,040
80 28,168 43,416 56,941 68,934 56,976 74,964 77,810 81,276 102,407 109,407
90 37,788 53,242 52,783 86,426 69,794 89,276 99,454 92,360 134,664 123,193
100 43,305 53,178 72,923 85,469 96,775 114,274 123,481 139,402 123,782 121,849
Problem Set #10
n/d 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%
20

1,804

4,383

5,663

5,314

9,060

10,530

10,187

12,421

11,443

9,886
30 9,245 7,948 11,053 11,037 12,295 16,708 25,427 20,089 23,164 19,871
40 9,294 16,404 12,457 18,970 32,203 30,651 17,987 21,592 28,036 30,528
50 14,116 27,620 18,586 30,031 40,716 30,978 38,709 36,832 41,240 50,161
60 16,208 25,331 41,537 31,088 42,787 36,178 45,494 70,937 66,018 53,866
70 25,421 30,101 37,891 48,993 52,143 49,537 52,725 81,628 87,709 90,223
80 34,706 48,068 55,561 60,129 70,699 78,746 94,287 101,349 101,557 92,824
90 47,012 51,118 61,811 70,020 76,960 93,918 87,436 111,408 128,928 98,643
100 45,696 53,786 78,050 89,040 73,232 105,027 109,588 120,721 136,048 126,335
 

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Last updated: May 20, 2004.