May 2025 - Challenge
Imagine the following scenario: You are entering an empty classroom. The board only says "Arithmetic in F_{?????}" from which you understand the class involved computations over the prime field F_{p} for some prime p, but you don't know which p.
It seems the students received their graded work, which lies on some of the tables. Each worksheet consists of the same multiple choice questions, and the grading marks the number of correct answers, without specifying which answers were correct.
It seems the questions, along with their possible answers, are:
5x_2 + 7x_3 + 5x_5 + 3x_6 = ? A: 2416 B: 2415 C: 2413 D: 2412 8x_1 + 5x_2 + 2x_5 + 5x_8 = ? A: 7624 B: 7622 C: 7621 D: 7625 7x_1 + 4x_2 + 5x_4 + 2x_5 = ? A: 638 B: 642 C: 633 D: 637 7x_1 + 9x_3 + x_6 + x_7 = ? A: 765 B: 761 C: 759 D: 760 8x_3 + 7x_4 + 4x_5 + 3x_6 = ? A: 2211 B: 2212 C: 2217 D: 2216 2x_3 + 5x_4 + 5x_6 + 5x_7 = ? A: 3497 B: 3501 C: 3496 D: 3499 5x_2 + 4x_4 + 6x_5 + 7x_6 = ? A: 4007 B: 4010 C: 4008 D: 4013 7x_3 + x_4 + 8x_6 + 8x_7 = ? A: 5531 B: 5535 C: 5538 D: 5534 x_1*x_2*x_3*x_4*x_5*x_6*x_7*x_8 = ? A: -4 B: 1 C: 5 D: 4 x_1*x_3*x_5*x_7 = ? A: 6544 B: 6546 C: 6545 D: 6550It seems all these questions deal with computations over F_{p} with eight unknown quantities, x_1,\ldots,x_8. Try as you might, you can't find any indications to the values of x_1,\ldots,x_8 anywhere; they must have been on the board and erased. Collecting the answers of the students along with their grade, you obtain
BABBDDCDCA 6/10 DADBDDAACD 6/10 CADBDBCDCC 7/10 DACBDDCBCB 6/10 DCBBDDCDAB 6/10 DDDBDDCDBD 8/10 DADBDDCCCC 8/10 CADBDDDDAC 7/10
Your goal: Find p.
A bonus "*" will be given for solving this followup question: After determining p, you encounter a second set of tests. This time, the board says "q(x) = x^{20}+???...?" so you understand you are dealing with a monic polynomial of the 20th degree, but most of it is erased.
The questions you find are nowq(5698) = ? A: 5146 B: 5142 C: 5149 D: 5147 q(1616) = ? A: 3715 B: 3725 C: 3720 D: 3716 q(1338) = ? A: 2547 B: 2548 C: 2549 D: 2550 q(7821) = ? A: 6624 B: 6628 C: 6619 D: 6627 q(4461) = ? A: 4661 B: 4660 C: 4659 D: 4662 q(2156) = ? A: 5112 B: 5107 C: 5106 D: 5102 q(7559) = ? A: 3916 B: 3921 C: 3913 D: 3917 q(5812) = ? A: 2340 B: 2342 C: 2345 D: 2344 q(794) = ? A: 6210 B: 6212 C: 6217 D: 6215 q(2595) = ? A: 3642 B: 3644 C: 3646 D: 3641 q(1640) = ? A: 1581 B: 1584 C: 1586 D: 1583 q(6779) = ? A: 6543 B: 6536 C: 6541 D: 6544 q(8362) = ? A: 3823 B: 3828 C: 3824 D: 3832 q(4605) = ? A: 5601 B: 5598 C: 5602 D: 5595 q(420) = ? A: 1272 B: 1274 C: 1278 D: 1275 q(8724) = ? A: 4410 B: 4420 C: 4415 D: 4412 q(3669) = ? A: 8974 B: 8973 C: 8976 D: 8971 q(1869) = ? A: 6723 B: 6720 C: 6728 D: 6721 q(7516) = ? A: 8244 B: 8247 C: 8246 D: 8243 q(1386) = ? A: 1571 B: 1570 C: 1569 D: 1568 q(4529) = ? A: 5504 B: 5501 C: 5503 D: 5506With answers
CCCDDCAADCCCBBDABDBAD 9/21 ADDACCADDDDBACDDBADCD 11/21 DCBABBAADDACBBAAAABDC 12/21 ACAAABABDDDBACAABADDC 11/21 CABAAAABDBDCCBDCBDBCC 12/21 DDDADCADDBDABBDBCADCC 12/21 ACDBBDAADDDABDCBCBCAC 9/21 AABCBDACDBADBDDCDACCB 10/21 BABBBBABAABCBDBCDACCA 9/21 ACCBABADDDDDBBCDDBACC 12/21 ACDBCBBDBDDCBCCADBCAA 9/21Your goal is to determine q(1).
We will post the names of those who submit a correct, original solution! If you don't want your name posted then please include such a statement in your submission!
We invite visitors to our website to submit an elegant solution. Send your submission to the ponder@il.ibm.com.
If you have any problems you think we might enjoy, please send them in. All replies should be sent to: ponder@il.ibm.com
Challenge:
30/04/2025 @ 12:35 AM EST
Solution:
09/06/2025 @ 12:00 AM EST
List Updated:
16/06/2025 @ 14:05 PM EST
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