Donate to Remove ads

Got a credit card? use our Credit Card & Finance Calculators

Thanks to johnstevens77,Bhoddhisatva,scotia,Anonymous,Cornytiv34, for Donating to support the site

Psi

cinelli
Lemon Slice
Posts: 550
Joined: November 9th, 2016, 11:33 am
Has thanked: 231 times
Been thanked: 160 times

Psi

#74971

Postby cinelli » August 16th, 2017, 9:29 pm

PSI       PSI
PI+ PIx
--- ----
RHO BETA

Cinelli

p.s. I have just noticed that the old Motley Fool discussion boards have gone. This seems a shame - all those wonderful puzzles and solutions never to be seen again.

UncleEbenezer
The full Lemon
Posts: 10691
Joined: November 4th, 2016, 8:17 pm
Has thanked: 1459 times
Been thanked: 2965 times

Re: Psi

#74991

Postby UncleEbenezer » August 16th, 2017, 11:25 pm

Inferring rules from previous posts, I take it:
- this is base10 arithmetic
- each letter represents a digit
- each digit 0-9 is represented exactly once
- 0 is disallowed as leading digit in any number
I think I avoid spoilers by matching font colour to background? OK, not reliable, but not so important as to matter.

First key to it is P, which we must be 1-3 to satisfy the rules (else BETA needs five digits).

From the addition, we have R = P+1 and P+S >= 10. This eliminates P=3, and from uniqueness of digits we can also eliminate P=1/S=9. So we know P=2, R=3, and S is 8 or 9.

Now we also have constraints on I: uniqueness under multiplication and addition eliminate 0, 1, 2, 5 or 6. So we're down to 8 cases to enumerate. Getting dumb,
294
24
---
318

294
24
----
7056

Gengulphus
Lemon Quarter
Posts: 4255
Joined: November 4th, 2016, 1:17 am
Been thanked: 2628 times

Re: Psi

#75161

Postby Gengulphus » August 17th, 2017, 2:59 pm

UncleEbenezer wrote:I think I avoid spoilers by matching font colour to background? OK, not reliable, but not so important as to matter.

Neat way of doing spoilers, yes - wish I'd thought of it! And it even poses a (fairly simple and multi-solution) puzzle, namely how one gets to see the spoiler when one wants to!

UncleEbenezer wrote:From the addition, we have R = P+1 and P+S >= 10.

No, you've forgotten to allow for a possible carry there, which means the number 10 needs to be replaced by 9, leading to more possibilities than you've seen.

Gengulphus

Gengulphus
Lemon Quarter
Posts: 4255
Joined: November 4th, 2016, 1:17 am
Been thanked: 2628 times

Re: Psi

#75293

Postby Gengulphus » August 18th, 2017, 12:44 am

cinelli wrote:
PSI       PSI
PI+ PIx
--- ----
RHO BETA


Using UncleEbenezer's spoiler technique...

As UncleEbenezer says, P must be in the range 1-3, as the product would have five digits otherwise, and the sum tells us that there must be a carry from the middle column to the left column, as otherwise P and R would be equal. So R = P+1, but since there might also be a carry from the right column to the middle column, we only know that S+P >= 9. We can also rule out I = 0, 1, 5 or 6 because those values would imply A=I (and in the case of 0, also that O=I).

If P=3, we know S >= 6 and I >= 2, and so can deduce that:

BETA = PSI * PI = (100P+10S+I) * (10P+I)
= 1000*P^2 + 100*(P*I+S*P) + 10*(S*I+I*P) + I^2
= 1000*P^2 + 100*P*(I+S) + 10*I*(S+P) + I^2
>= 1000*9 + 100*3*8 +10*2*9 + 4
= 9000 + 2400 + 180 + 4
= 11584

which is incompatible with BETA being a 4-digit number. So P=3 is not possible.

If P=1, then R=2 and S can only be 8 or 9. The product must be > 10*100 = 1000 and < 20*200 = 4000, so B must be 1, 2 or 3, and since 1 and 2 are already taken, it must be 3. The middle column of the sum (including any carry from the right column) must add up to 10 or 11, implying H=0 or 1, and since 1 is already taken, H must be 0 and the middle column must add up to 10. It can do that as S+P+carry = 9+1+0 or 8+1+1. In the first case, I must be <= 4 to avoid generating a carry into the middle column, but that implies BETA = PSI * PI < 200 * 15 = 3000, which is incompatible with B=3.

So S=8 and there is a carry from the right column to the middle column, i.e. I >= 5. We've already ruled out I=5 or 6, and I cannot be 8 (since it duplicates S), nor 9 (since that would imply O=8, duplicating S). So I=7 and the addition is 187+17 = 204. But that means the multiplication is 187*17 = 3179, duplicating 1 and 7. So P=1 doesn't work.

Having ruled out P=1 and P=3, we can deduce that P=2, R=3 and S can only be 7, 8 or 9. We've already ruled out values of 0, 1, 5 and 6 for I, and that rules out 2 and 3 as well, so I can only be 4, 7, 8 or 9. If I is 7, 8 or 9, then using the fact that I and S must be different, I+S must be 15, 16 or 17, and so BETA = 1000*P^2 + 100*P*(I+S) + 10*I*(S+P) + I^2 > 1000*P^2 + 100*P*(I+S) >= 1000*4 + 100*2*15 = 7000. So B is also one of 7, 8 or 9, and S, I and B account for all three of 7, 8 and 9 between them. But now I=7 is impossible because the product implies that A=9, and I=9 is impossible because the sum implies that O=8. So I=8, which implies that O=6 and there is a carry from the right column of the addition into the middle column. That allows us to rule out S=9, since the middle column would then sum to S+P+carry = 9+2+1 = 12, implying H=2 when 2 has already been used. So S=7, implying that the sum is 278+28 = 306 and the product 278*28 = 7784 when 7 has already been used.

So we're left with the possibility I=4, implying that O=8 and there isn't a carry from the right column of the addition into the middle column. That disallows S=7 because it makes the middle column S+P+carry = 7+2+0 = 9, not generating a carry into the left column, and it disallows S=8 because that would duplicate 8. So S=9, the addition is 294+24 = 318, and as a check that there is indeed a solution, the product is 294*24 = 7056, which uses the four remaining digits.


Gengulphus

cinelli
Lemon Slice
Posts: 550
Joined: November 9th, 2016, 11:33 am
Has thanked: 231 times
Been thanked: 160 times

Re: Psi

#75584

Postby cinelli » August 18th, 2017, 9:17 pm

Well solved. And the hiding of the solution is a master stroke.

Cinelli


Return to “Games, Puzzles and Riddles”

Who is online

Users browsing this forum: No registered users and 3 guests