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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

4,896

7,219

14,239

14,597

17,067

17,211

18,880

17,218

26,705

31,616
30 8,681 13,322 17,664 29,178 23,555 30,429 36,623 43,077 39,684 47,037
40 20,263 34,226 43,873 39,601 48,033 55,416 61,932 44,651 74,426 75,797
50 24,015 40,902 64,958 69,484 54,033 67,269 87,727 97,185 67,964 77,126
60 36,415 47,598 60,796 56,823 60,899 108,333 118,023 144,730 110,513 156,487
70 39,613 87,197 76,526 71,589 97,032 132,993 142,052 112,603 168,204 188,446
80 80,079 84,379 137,177 128,113 152,768 146,718 162,836 165,853 211,358 170,994
90 61,569 103,941 114,372 154,224 174,975 182,090 197,562 189,095 254,423 216,724
100 61,492 120,519 155,225 177,956 195,310 249,526 206,565 229,485 224,542 297,172
Problem Set #2
n/d 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%
20

4,239

10,276

10,944

12,707

11,743

17,489

15,872

20,854

18,968

29,716
30 14,134 19,051 18,785 24,263 24,128 28,440 32,089 37,878 39,783 41,067
40 25,414 19,638 45,041 52,391 30,488 39,052 62,086 66,257 64,861 64,941
50 22,801 45,137 45,345 49,521 68,757 80,033 89,049 75,741 118,480 113,467
60 39,890 53,055 61,735 71,270 124,227 83,610 106,879 98,834 124,532 110,279
70 56,265 71,499 76,728 87,832 111,624 113,307 126,457 135,876 143,509 181,684
80 47,853 73,107 116,079 99,683 164,403 124,087 132,873 165,590 159,625 204,879
90 84,659 91,162 127,290 171,470 168,937 216,552 188,819 207,400 241,699 279,535
100 81,939 126,340 156,222 167,522 207,466 227,994 228,198 228,515 250,623 253,717
Problem Set #3
n/d 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%
20

10,770

7,042

14,559

12,268

13,452

20,906

13,858

16,658

17,761

23,708
30 10,462 17,340 22,428 32,921 35,850 32,077 50,675 41,296 46,535 42,691
40 23,646 32,577 33,451 39,964 40,746 46,289 44,748 53,503 52,058 58,027
50 24,717 35,077 55,782 43,437 66,278 76,446 56,810 100,106 92,066 82,225
60 44,889 55,359 75,997 76,344 89,942 83,523 123,668 132,767 131,997 125,442
70 49,107 70,730 91,440 82,852 108,767 110,166 149,561 123,270 129,109 145,630
80 53,843 86,595 85,133 117,017 124,014 155,478 175,972 176,989 188,048 213,017
90 80,716 86,464 147,735 132,395 150,243 194,504 188,797 192,102 212,972 229,593
100 78,424 122,530 146,620 173,629 169,886 223,687 201,749 219,405 272,604 276,835
Problem Set #4
n/d 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%
20

4,209

9,613

15,190

17,681

11,619

11,458

23,858

16,783

21,082

27,402
30 14,070 19,423 17,614 22,968 23,819 44,714 31,226 39,131 48,522 33,497
40 22,185 32,231 26,722 42,965 47,827 42,372 50,184 62,312 57,915 64,148
50 34,690 47,538 42,200 58,345 84,293 74,000 79,645 102,063 90,467 101,676
60 53,302 71,613 55,399 83,770 82,329 99,693 111,519 92,751 126,846 138,929
70 54,918 76,945 84,212 107,696 123,603 136,274 158,181 173,801 147,231 146,806
80 62,166 76,015 104,389 115,926 115,752 173,651 118,016 182,924 159,714 190,985
90 75,445 106,879 133,486 145,363 167,047 163,693 145,265 208,398 254,421 267,996
100 72,354 110,694 150,299 191,031 157,426 187,100 209,660 233,244 275,832 300,768
Problem Set #5
n/d 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%
20

6,434

14,519

10,199

13,243

12,988

18,513

24,396

22,014

26,975

21,976
30 9,599 19,601 13,205 19,691 42,270 29,230 36,438 37,344 29,516 38,540
40 21,586 32,058 31,087 38,289 47,433 42,896 52,049 68,318 57,145 77,397
50 29,662 36,728 46,735 56,581 85,459 50,221 64,162 75,228 97,895 89,172
60 44,947 53,393 79,274 68,219 76,103 105,276 86,511 105,124 148,903 121,360
70 40,730 58,855 94,364 113,027 137,241 118,565 127,260 160,912 160,500 185,005
80 63,688 74,213 100,246 125,813 137,391 151,225 162,712 165,751 180,131 220,276
90 75,234 104,395 115,315 161,462 146,194 174,924 200,457 200,958 204,164 233,829
100 79,772 121,667 110,371 180,962 223,641 228,527 209,600 245,877 276,721 306,291
Problem Set #6
n/d 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%
20

7,702

12,208

8,317

10,226

9,439

15,206

19,140

16,542

18,016

23,622
30 11,328 12,942 19,613 22,846 38,523 33,520 45,040 52,097 42,424 34,749
40 26,885 43,288 33,022 34,577 36,896 53,431 53,694 64,834 73,362 52,875
50 18,707 39,859 48,468 54,488 55,021 72,546 66,480 80,854 66,515 89,897
60 40,024 45,795 90,846 61,538 74,318 114,191 116,769 119,560 135,528 96,408
70 55,459 81,286 88,588 98,384 115,226