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# extremely hard number theory

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Hi. I’ve been stuck on this problem for months already. My friend asked me this question, expecting me to answer but I am stuck on it too. I looked and looked… but there doesn’t seem to be an answer that flows. Any hints as to where to start on this tough number theory problem? Thanks, - Jimmy.

The problem goes like this: Sherlock Holmes and Dr. Watson recover a suitcase with a three-digit combination lock from a mathematician turned criminal. Embedded in the suitcase above the lock is the cryptic message "AT SEA BASE. SEAS EBB SEA: BASS. "

Dr. Watson comments, "This probably isn't about ocean fish. Perhaps it is an encrypted message. The colon suggests the addition problem , where each letter represents a distinct digit, and the word 'BASE' implies that the problem is in a different base."

Holmes calmly turns the combination lock and opens the suitcase. While Dr. Watson gapes in surprise, Holmes replies, "You were indeed right, and the answer was just the value of the word , interpreted as decimal digits." What was the lock combination?

May 13, 2021

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code is 871

trust me i just did this and got it right

May 13, 2021
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I'm not looking for the answer. I'm looking for a well approach that can be used in similar problems. That way I learn. If I just scavenge the answer, do I learn?

MathProblemSolver101  May 15, 2021
#2
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At sea base

SEAS + EBB + SEA = BASS

Let's call the base c cause it's the sea. :))

(S)c^3 + (E + E + S)c^2 + (A + B + E)c + S + B + A = (B)c^3 + Ac^2 + Sc + S

(S)c^3 + (E + E + S)c^2 + (A + B + E)c + B + A = (B)c^3 + Ac^2 + Sc

B + A = c, because the sum up to 0 (mod c), and both can't be 0.

That would mean that there would be a carry 1.

A + B + E + 1 = S (mod c)

c + E + 1 = S (mod c)

E + 1 = S

E + E + S + 1 = A (mod c)

This is where I had to do a bit of guessing. I felt like E + E + S + 1 = A would be a carry 2.

I'm sure you could test for carry 1 and carry 2 and see if they both work.

2 + S = B

So what are our equations?

B + A = c

E + 1 = S

E + E + S + 1 = A + 2c

2 + S = B

Now, time for solving. I don't actually have a good method for this.. I just tried everything until I got something. *Any tips for solving equations like these would be helpful.

Looking at my scratch paper, I don't think I even know how I solved it.

A = 1

B = 10

E = 7

S = 8

c = 11

It asks for a 3 digit number. At first, I wasn't sure which word to use, but since the word has to be 2 - 3 letters, and AT and EBB doesn't work, it must be SEA.

SEA = 871

Hopefully that's correct. :))

=^._.^=

May 13, 2021
#3
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Thank you so much! I've been doing some math and equations but always ended up with a contradictory statement... and so was stuck on this for quite a while. I really appreciate your help, as I am, relatively abysmal in number theory compared to you. Jimmy

MathProblemSolver101  May 13, 2021
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Thank you for the compliment. :3

Also, I've seen your past answers, you're super talented at math, especially with geometry and probability.

=^._.^=

catmg  May 13, 2021