[rookie1ja]

5-digit Number - Back to the Number Puzzles
What 5-digit number has the following property? If we put numeral 1 in front of the number, we get a number three times smaller, than if we put the numeral 1 behind this number.

Spoiler for Solution:

5-Digit Number - solution
Using an easy equation: 3(x+100000) = 10x+1 we find out that the number is 42857.

[cementheads]

10007

[earl11]

42857

[sql_er]

There is actually another way to solve this puzzle, figuiring out the answer digit by digit, backwards.

Can someone figure out what it is?

[zAlbee]

Quote

There is actually another way to solve this puzzle, figuiring out the answer digit by digit, backwards.

Can someone figure out what it is?

Yes, I did it this way. It is slower, but perhaps more fun.
First you write the equation like a long multiplication question:

1 a b c d e
x_______3
a b c d e 1

This is the logic:

e x 3 ends in 1. e must be 7 (7x3=21, carry the 2)
d x 3 ends in 7-2. d must be 5 (5x3=15, carry the 1)
c x 3 ends in 5-1. c must be 8 (8x3=24, carry the 2)
b x 3 ends in 8-2. b must be 2 (2x3=6, no carry)
a x 3 ends in 2. a must be 4 (4x3=12, carry the 1)
1 x 3 ends in 4-1. this is true.


______2
1 a b c d 7
x_______3
a b c d 7 1

_____1
1 a b c 5 7
x_______3
a b c 5 7 1

___2
1 a b 8 5 7
x_______3
a b 8 5 7 1


1 a 2 8 5 7
x_______3
a 2 8 5 7 1

1
1 4 2 8 5 7
x_______3
4 2 8 5 7 1

(carries in yellow)

[sql_er]

Great job!

[slmo]

I solved algebraically but I had to see zAlbee's way for myself...and it works! Very clever solution

[Braindead_Genius]

The original solution doesn't work. It gives you a decimal that, when rounded, gives you the correct answer. But, if you change it to -1 it removes the decimal. Just pointing out that it needs a slight adjustment. It took me awhile and I resorted to guess and check but, thanks to my dad's idea, I used microsoft excel.

[spoxjox]

Quote

The original solution doesn't work. It gives you a decimal that, when rounded, gives you the correct answer.

No, try it again. The original solution works fine. 100,000 + x = (10x + 1) / 3 resolves uniquely to x = 42,857.

[IceCreamTruck]

Very cleaver guys. I tried to not use a calculator, so I made it a little harder for me. I assumed the answer was "55555" to start and did the math and it was close but not right.

Then I tried "33333" and that was close and not right, but it showed me the answer was between the two.

Then I tried "44444" and that was closer and then I knew the answer was between 44444 and 33333.

I got into the right ballpark, but I cheated and looked at this point. At least I narrowed to 11111 possible answers.

which, 11111, is consequently the only 5 digit number thats solution is exactly the same when you ad a 1 to the beginning or end. LAUGH!!!

This post has been edited by IceCreamTruck: 12 February 2008 - 03:22 PM