Use pure logic and deductive reasoning only. Document detailed steps in arriving at final solution.

You can use calculator minimally.

$ Given$:

Utilizing These two unique set of Pan digital fractional relations, solve for all the digits. You can try solving first without helping hand from this U V C equation

U, V, C are 3 distinct digits..values can vary from 1 to 9.

CU is a concatenated number

$U^V x V^U$ = $CU$

$Unique $Set of $Pan digital $ $fractional $ and $Decimal Relationships:$

All the letters represent distinct digits(1 to 9).

All the words in both equations on either side represent concatenated Numbers




1 Answer 1


$U = 2$ $V = 3$ $C=7$ Reasoning
$u^v \times v^u = cu$ means that the multiplication must result in a 2 digit number.
This means u and v are small digits and there are not so many cases. We can try them out.
None of them can be 1 as the result it would be a 1 digit number.
u=2, v=3 results in 8x9 = 72. Good
u=2, v=4 results in 16x16>100. Stop incrementing v
u=3, v=2 results in 9x8 = 72. Not good.
Rest of the cases go over 100.

From the second division:

I actually transformed it into a multiplication and concluded that a can be 0 or 5 otherwise we don't get a n integer from 𝑂𝐿𝐴𝐺𝐸𝑉𝐡. But A cannot be 0 since the first division contains one factor that starts with A. SO A=5.


From the same second division, then decimals are now o75. This, multiplied by 72 must be divisible with 1000 in order to get an integer. The only possible values for o are 3 and 8, but 3 is already taken. So o=8.
From the same equation I got that b=1 since a six digit number (ignore decimals) multiplied by 72 we get a 7 digit number. if b>=2 we get an 8 digit number.
Taking decimals again we get 875*72 = 63000. This means that 2*L+3 = 1 (mod 10). So 2*l = 8 (mod 10). So L can be 9 or 4.
but bugvel.oca = 12g3el.875 > 120000. And 120000*72 = 864000 so 𝑂𝐿𝐴𝐺𝐸𝑉𝐡 > 864000. So L>6. So L=9.

Now we are left with...

2 letters g,e and 2 digits. 4,6. We could probably calculate them but there are only 2 combinations. So let's do the math.
g=4, e = 6: 124369,875Γ—72 = 8954631. Valid.
g=6 e=4: 126349,875Γ—72 = 9097191. Not valid.

Now we got all:

B=1, U=2, V=3, G=4, A=5, E=6, C=7, O=8, L=9.

  • $\begingroup$ Excellent deductive quick work...would you be able to solve it without Uvc relation? $\endgroup$
    – Uvc
    May 24, 2019 at 15:24
  • $\begingroup$ Not sure. I didn't even tiuch one of the divisions. I didnt even check it because i am lazy. Maybe, maybe not $\endgroup$
    – Marius
    May 24, 2019 at 16:48

Your Answer

By clicking β€œPost Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.