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# Senior maths challenge question watch

1. I came across this question while preparing for the senior team maths challenge in a few days, can anyone give me a hand?

The question is:

"Given any three-digit number,, define to be: x- the sum of the squares of the digits of x.
What is the maximum possible value of ?"

I got as far as defining where the number x is written abc. However I don't know where to go from there, does it involve differentiating of some sort? Anyway any help would be appreciated.
2. Hint: Maximize and others separately.

Spoiler:
Show
Complex the square

Differentiating could be used, but it's less "nice"

PS. Good luck, and congratulations for beating Jersey
3. For "less "nice"" read "cambridge standard"
4. Less elementary
5. Sorry if I'm missing something silly here, but wouldn't the solution just be 9^2 + 9^2 + 9^2 = 243?
6. (Original post by marcusmerehay)
Sorry if I'm missing something silly here, but wouldn't the solution just be 9^2 + 9^2 + 9^2 = 243?
you are missing something silly here :P
7. its x minus the sum of the squares
8. Wow, I need my eyes testing.

EDIT: I think I know what the answer should be just from a small amount of logic rather than any extensive calculation.

Spoiler:
Show
951?
9. Does anyone agree with my answer?

Spoiler:
Show
951

EDIT: I agree marcus
10. (Original post by Dominorum)
Does anyone agree with my answer?

Spoiler:
Show
951

EDIT: I agree marcus
Agreed, although x could also be 950. Hence, the answer is A(951) (which is equal to A(950)).
11. one way to do it is to consider each digit separately in your equation (as simonm suggested), and differentiate (as simonm advised against )
12. (Original post by Dominorum)
Does anyone agree with my answer?

Spoiler:
Show
951

EDIT: I agree marcus
yep :>

i found it quicky by just quickly checking each digit, but a more systematic approach:

maximise 100a - a^2, i.e. 2a = 100, a = 50

so pick the closest digit you can to 50, i.e. 9

maximise 10b - b^2, i.e. 2b = 10, b = 5

so pick 5

maximise c - c^2, i.e. 2c = 1, c = 1/2

so pick 0 or 1
13. ok i think this is the right answer (where

Max value of occurs at are all at a maximum

Max value of occurs when technically but this is not in the given range of values for - sketching will show is the max value of

Max value of occurs when
( At max when

sketching again will show max value of is at max when or

when
when
when or

So Max value of
14. (Original post by marcusmerehay)
Wow, I need my eyes testing.

EDIT: I think I know what the answer should be just from a small amount of logic rather than any extensive calculation.

Spoiler:
Show
951?
Yes the number could either be 951 of 950 but your question asked for max value of A(x) which is 844

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Updated: January 29, 2013
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