SOLV for Narcissistic UVC

This puzzle highlights some cyclic power Relations.

Please provide detailed reasoning of derivation of digits from various power relations given below:

$$Given$$:

$$A$$, $$C$$, $$L$$, $$N$$, $$O$$, $$S$$, $$U$$, $$V$$ are all distinct digits varying from 0 to 9.

$$SOLV$$, $$UVC$$, $$LSU$$, $$VAN$$ are concatenated Numbers.

1) $$S^L$$ + $$O^L$$ + $$L^L$$ + $$V^L$$ = $$UVC$$

2) $$U^L$$ + $$V^L$$ + $$C^L$$ = $$LSU$$

3) $$L^L$$ + $$S^L$$ + $$U^L$$ = $$VAN$$

4) $$V^L$$ + $$A^L$$ + $$N^L$$ = $$UVC$$

Back to Step 2..Repeat and Rinse Cycle of nice Power Relations.

• Don’t be scared of number of equations. Deductive logic and reasonable assumptions will get you to the right result. This is no more difficult than some of my previous ones. – Uvc Jun 14 at 10:43

I believe the following values are consistent with the equations:

A = 6, C = 7, L = 3, N = 0, O = 4, S = 5, U = 2, V = 1

My first discovery is that

If L is 4 or greater, one of the remaining digits must be 6 or greater since there are eight of them total, and 6^4 exceeds 1000. Meanwhile, if L is 2 or less, this also does not work because of the second equation, where U^2+V^2+C^2 = 200 + SU. Assuming maximum values for U, V, and C, you would get 81+64+49 = 194, which is insufficient. Therefore, L must be equal to 3.

Working from Equations 1 and 4:

S^3+O^3+3^3 = A^3+N^3, as both equations equal to UVC and the V^L's cancel out. I feel like there may have been a more elegant way to figure this out, but I recall that 3^3 + 4^3 + 5^3 = 6^3, so I believe S and O are 4 and 5 while A and N are 0 and 6.

This gives us:

216 + V^3 = UVC The remaining possibilities for V are 2, 1, 7, 8, and 9, however we can rule out the latter two from equation 2 as 8^3 exceeds 400 and therefore cannot sum to LSU. 7 does not work as you result with 559, therefore V would have to be both 5 and 7. 2 does not work as your result would be 224, and both C and S/O cannot be 4. Lastly V=1 gives you a result of 217, making U = 2, V = 1, and C = 7. So far we are consistent.

Now back to equation 2:

We have 2^3 + 1^3 + 7^3 = LSU, resulting in LSU = 352, giving us L = 3 and U = 2, which is consistent, and S = 5, leaving O to equal 4. We are now left with A and N.

Finally with equation 3:

3^3 + 5^3 + 2^3 = VAN, resulting in VAN = 160. Once again confirming V = 1, as well as A = 6 and N = 0, consistent with our original hypothesis from Equations 1 and 4.

• Excellent deductive reasoning leading to the right result..this is one of my favorite..most of the people are put off when they see bunch of equations – Uvc Jun 19 at 22:21