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# Six integer equations, Eight digits

Best Answer Prime, 25 January 2013 - 07:34 AM

I found couple solutions, but not with as many digits, I'm afraid.

Spoiler for 7-digit solution
Go to the full post

9 replies to this topic

### #1 bonanova

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Posted 24 January 2013 - 08:47 AM

Edit. Removed misleading information from the original post.

I ran across this puzzle - it's part of a larger puzzle - a week ago.

There are twelve letters, each of which corresponds to one of eight digits.

The letters are juxtaposed into two-digit numbers, and they occur that way in six equations.

Here are the equations - six, in twelve unknowns.

Their integer properties provide additional constraints.

VG = CV - QP
JY = GP - ZY
QG = VJ - MV
XC = BC - XP
XK = KX - QB
ZP = CY - GM

Can you find consistent values of the 12 letters B C G J K M P Q V X Y Z

using just eight digits?

Edited by bonanova, 24 January 2013 - 11:11 AM.
Remove incorrect information from original post

• 0
The greatest challenge to any thinker is stating the problem in a way that will allow a solution.
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### #2 superprismatic

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Posted 24 January 2013 - 08:24 PM

Please check the equations for typos.  I wrote a program to look for solutions and it found none.

I looked for bugs to no avail.  It is certainly is easier to check the equations for typos than to find

an easily overlooked bug.

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### #3 Prime

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Posted 24 January 2013 - 08:35 PM

Edit. Removed misleading information from the original post.

I ran across this puzzle - it's part of a larger puzzle - a week ago.

There are twelve letters, each of which corresponds to one of eight digits.

The letters are juxtaposed into two-digit numbers, and they occur that way in six equations.

Here are the equations - six, in twelve unknowns.

Their integer properties provide additional constraints.

VG = CV - QP
JY = GP - ZY
QG = VJ - MV
XC = BC - XP
XK = KX - QB
ZP = CY - GM

Can you find consistent values of the 12 letters B C G J K M P Q V X Y Z

using just eight digits?

Can we assume, there are no leading zeroes in any of those 2-digit numbers?

• 0

Past prime, actually.

### #4 jddouglas

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Posted 24 January 2013 - 10:26 PM

When I tried to work this out, I hit a dead end pretty quickly. Makes me think this isn't solvable. Maybe I'm missing something obvious...

Spoiler for

• 0

### #5 BobbyGo

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Posted 24 January 2013 - 11:06 PM

Spoiler for jddouglas

Spoiler for G

Sorry, had trouble with the spoilers.

Edited by BobbyGo, 24 January 2013 - 11:09 PM.

• 0

### #6 Prime

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Posted 25 January 2013 - 07:34 AM   Best Answer

I found couple solutions, but not with as many digits, I'm afraid.

Spoiler for 7-digit solution

Edited by Prime, 25 January 2013 - 07:36 AM.

• 0

Past prime, actually.

### #7 Prime

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Posted 25 January 2013 - 07:50 AM

Now that I took few minutes (enough to prohibit editing previous post) to study the statement of the problem closely, I find my solution (above) answers OP's requirements in full.

• 0

Past prime, actually.

### #8 bonanova

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Posted 25 January 2013 - 08:21 AM

Now that I took few minutes (enough to prohibit editing previous post) to study the statement of the problem closely, I find my solution (above) answers OP's requirements in full.

Yes. It does. Great solve.

To clarify, the OP might better have read: Each letter has a value selected from just eight digits.

The actual wording, and the title, suggests (wrongly) that eight digits are used.
• 0
The greatest challenge to any thinker is stating the problem in a way that will allow a solution.
- Bertrand Russell

### #9 CaptainEd

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Posted 25 January 2013 - 04:19 PM

Nothing wrong with the wording. You didn't give away the trick, but you didn't hide it. It's just that many of us (*palm*) weren't open minded. I, too, couldn't get anything but all zeroes.

Good puzzle! Thanks again for the education!

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### #10 Prime

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Posted 25 January 2013 - 09:52 PM

I like the original wording. It is fair and accurate. It does not actively mislead, rather allows solvers to mislead themselves. It could use no leading zeroes stipulation to prohibit all zero solution, but then it would simplify solving process quite a bit.

The fact that some people suspected an error in the problem statement, while Bonanova remained silent on the subject served as a hint to me.

The title could be “6 equations, 12 unknowns,” or even more catchy “6 against 12,” to attract more attention.

Great puzzle!

• 0

Past prime, actually.

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