Unknown Digits

A forum for good logic/math puzzles.

Moderators: jestingrabbit, Moderators General, Prelates

newbie11
Posts: 12
Joined: Fri Oct 05, 2012 1:09 am UTC

Unknown Digits

Postby newbie11 » Thu Feb 11, 2016 1:50 am UTC

Below is a simple logic problem with 2 x options, you will know you are right because both should give you the same answers. Let the first digit be D1, the second digit be D2 etc.

OPTION 1
Q = D2 + D4
R = D5 + D2 + 1
S = D1 - D3 + D2
T = D4 + D5 + 1
U = D1 - D5
V = D3 - D5 + D2
W = D4 + 1
X = D3 + D5

OPTION 2
Q = D5 - D4 - D3
R = D3 + D4
S = D3 x 2
T = D1 + D4
U = D5 - D3 - D2
V = (D2 x D4) + D2
W = D2 + D1
X = (D3 x D4) + D2

newbie11
Posts: 12
Joined: Fri Oct 05, 2012 1:09 am UTC

Re: Unknown Digits

Postby newbie11 » Thu Feb 11, 2016 8:29 pm UTC

I should have mentioned that the digits D1, D2, D3, D4, D5 have to be a unique digit between 0-9.

mfb
Posts: 947
Joined: Thu Jan 08, 2009 7:48 pm UTC

Re: Unknown Digits

Postby mfb » Thu Feb 11, 2016 10:30 pm UTC

Are there any constraints on the values of Q, R and so on?

User avatar
ConMan
Shepherd's Pie?
Posts: 1671
Joined: Tue Jan 01, 2008 11:56 am UTC
Location: Beacon Alpha

Re: Unknown Digits

Postby ConMan » Fri Feb 12, 2016 1:14 am UTC

For that matter, are Q, R, etc the same in the two options? Because I'm not sure there's a consistent solution if they are.
pollywog wrote:
Wikihow wrote:* Smile a lot! Give a gay girl a knowing "Hey, I'm a lesbian too!" smile.
I want to learn this smile, perfect it, and then go around smiling at lesbians and freaking them out.

User avatar
emlightened
Posts: 42
Joined: Sat Sep 26, 2015 9:35 pm UTC
Location: Somewhere cosy.

Re: Unknown Digits

Postby emlightened » Fri Feb 12, 2016 12:33 pm UTC

Spoiler:

Code: Select all

for k in map(lambda s: '{:0>5}'.format(s), range(10**5)):
   D1,D2,D3,D4,D5=map(int,k)
   if all([
   D5 - D4 - D3 == D2 + D4,
   D3 + D4 == D5 + D2 + 1,
   D3 * 2 == D1 - D3 + D2,
   D1 + D4 == D4 + D5 + 1,
   D5 - D3 - D2 == D1 - D5,
   (D2 * D4) + D2 == D3 - D5 + D2,
   D2 + D1 == D4 + 1,
   (D3 * D4) + D2 == D3 + D5
   ]): print(k)


Returns nothing.

ConMan's right.
← approximately how gay I am.
"a nebulous cloud of different combinations of styling bodily features"
"on either side of a wrought iron fence made of tigers"

curiosityspoon
Posts: 35
Joined: Wed Sep 24, 2014 5:01 pm UTC

Re: Unknown Digits

Postby curiosityspoon » Sat Feb 13, 2016 10:42 am UTC

Since apparently there's supposed to be some kind of logic (?) from which actual deductions can be drawn, it's rather easy to show that the letters can't have any kind of consistency to them across options, without appealing to a brute force printout.

The two expressions that each represent T are "D4 + D5 + 1" and "D1 + D4". By the transitivity of equality, those expressions are equal to each other, and so D5 + 1 = D1. (Coincidentally, it's immediately apparent that U = 1 from this and its first expression, but we don't need to use that fact for anything.)

The two expressions that each represent W are "D4 + 1" and "D2 + D1". These are likewise equal, and D1, D2, and D4 (as well as D5) are distinct single digits.
If D2 = 0, then D4 + 1 = D1...but D5 + 1 is also equal to D1, so D4 is clearly equal to D5 in this case, violating the uniqueness of the D-variables.
If D2 = 1, then D4 will be equal to D1, again violating uniqueness of digits.
With those two values ruled out, no matter what digit D2 takes on, it will necessarily be the case that D4 > D1, and of course D4 > D5 as well.

The two expressions that each represent Q are "D2 + D4" and "D5 - D4 - D3", and are equal.
The first expression, as the sum of two distinct single digits, is clearly positive.
The second expression begins with D5 - D4, but since we've established that D4 must be greater than D5, that difference is negative. Further subtracting another single digit in D3 can only drive the expression further into the negatives, so the two expressions for Q cannot, in fact, be equal. Q.....ED, I guess.

Cauchy
Posts: 602
Joined: Wed Mar 28, 2007 1:43 pm UTC

Re: Unknown Digits

Postby Cauchy » Mon Feb 15, 2016 12:12 am UTC

curiosityspoon wrote:(Coincidentally, it's immediately apparent that U = 1 from this and its first expression, but we don't need to use that fact for anything.)


We can get to a faster contradiction using it. Since U = 1, D5 - D3 - D2 = 1, and so D5 = D2 + D3 + 1. Setting the two expressions for Q equal to each other, we get D2 + D4 = D5 - D4 - D3, or D2 + D4 = (D2 + D3 + 1) - D4 - D3, which simplifies to D4 = 1/2. This isn't a digit, obviously.

Dropping the requirement that the D's be digits, we find that setting the expressions for Q, R, S, T, and U equal to each other gives a unique solution for the D's:
D1 = 3/2
D2 = -3/4
D3 = 1/4
D4 = 1/2
D5 = 1/2

This isn't consistent with the expressions for V being equal to each other though.
(∫|p|2)(∫|q|2) ≥ (∫|pq|)2
Thanks, skeptical scientist, for knowing symbols and giving them to me.


Return to “Logic Puzzles”

Who is online

Users browsing this forum: No registered users and 7 guests