Question

Two of the three missing digits of the five-digit perfect square number 3_ ,__ 9 are the same. What is the other missing digit?
MEDAL

Answer 1

This question is from MathCounts.

Answer 2

COME ON SOMEONE ANSWER MY QUESTION. THE ANSWER IS 0.

Answer 3

117^2 = 31329

Answer 4

2 of the numbers need to be the same. READ.

Answer 5

try again :)
197^2 = 38809

Answer 6

yes thats right how to get it?

Answer 7

just trials

Answer 8

This is mathcounts You have to do it fast. I need to know how to do it. How do you get it? with 2 digits the same?

Answer 9

first, i use 200^2 get 40,000. ths closed but out of range
look the unit number is 9, so my guess the units must be 3 or 7
finally, i take 197

Answer 10

i still dont understand.

Answer 11

@whpalmer4 , @mathmale , @ganeshie8

they are the good explainers

Answer 12

ok thanks

Answer 13

hello whpalmer4 thanks

Answer 14

I think @RadEn covered it pretty well. 200^2 = 40000, so that's about spot where you need to start looking. If it is a perfect square ending in 9, you're only going to get that if the number being squared ends in a digit where its square ends in 9.

1*1 = 1 no good
2*2 = 4 no good
3*3 = 9 good
4*4 = 16 no good
5*5 = 25 no good
6*6 = 36 no good
7*7 = 49 good
8*8 = 64 no good
9*9 = 81 no good
0*0 = 0 no good

So, you might reasonably guess that the nearest numbers to 200 (from below) ending in 3 or 7 are your obvious candidates, which would be 193 and 197.

You can speed up the computation a bit, perhaps, if you remember that \[(a-b)(a-b) = a^2-2ab+b^2\]Let \(a=200\) and you have \(40000-800b+b^2\) and now you just plug in \(b=3\) or \(b=7\) to test \(197\) or \(193\) respectively.

Answer 15

can you guys please help me on my problem?

Answer 16

And looking at the \(40000-800b\) part you can immediately rule out \(b=7\) because the value will be \(<35000\).

Answer 17

@DavidUsa is there another problem I'm not seeing here?

Answer 18

Actually, I didn't look carefully enough at the original problem, you can't immediately rule out \(b=7\) because we don't know the second digit.

Answer 19

the only way I see how to do it is by trial and error.
as people have pointed out
the upper bound is 200
the lower bound is sqr(3)*100 or about 174
the unit digit must be either a 3 or 7. we have the following candidates:
177
183
187
193
197

not hard to check with a calculator.

Answer 20

only 197^2 gives a repeated number : 38809 the "unrepeated" number is 0

Answer 21

Doing it by hand, I'd start from the largest candidates first, simply because they are a bit quicker to calculate with my suggestion. Just a happy accident that you encounter the solution first!

Answer 22

or being "rainman" could be helpful

Answer 23

There's an entertaining Feynman story about mental calculation of cube roots here:
http://www.ee.ryerson.ca/~elf/abacus/feynman.html