Brain Teasers
You Could Go Infinite
What three-digit number when doubled is, by value, one more than the value of the same three digits when reversed?
If this is too easy, go for extra points included with hint.
If this is too easy, go for extra points included with hint.
Hint
A wrongheaded guess would say 123, because 123 * 2 = 321 + 1. The title hints you can easily go on to find for a ten-digit number that otherwise has same characteristics.Answer
Answer-397.
397 * 2 = 794 = 793 + 1.
Solution-
With lowercase letters standing for unknown digits, the problem is formulized as:
(100a + 10b + c) * 2 = (100c + 10b + a) + 1.
200a + 20b + 2c = a + 10b + 100c + 1.
199a + 10b - 98c - 1 = 0.
[The negative values achieve their negative extreme when c = 9. - 98c - 1 = -883. This caps the value of a as the integral portion of 883/199, i.e., 4. This is in brackets, because there are stronger forces which make this unimportant.]
Rule 1) No number starts with a 0. [a is not 0. c is not 0.]
Rule 2) Because doubling a number starting with 5, 6, 7, 8, or 9 will result in a product with an additional position/digit, a must be less than 5. [a is 1, 2, 3, or 4.]
Rule 3) When a digit is doubled, the most that can be "carried" to the next position is 1. This happens when and only when the digit doubled is 5, 6, 7, 8, or 9.
Rule 4) Any multiple of 2 will be an even number. To differ from the doubled number by just 1, the reversed number must be odd and its last digit must be odd. a must be an odd digit. [a is 1 or 3.]
Rule 5) Doubling a three-digit number results in a number determined by the leading digit and limited to two hundred possibilities. (Doubling a number in the 100's has a result in the 200's or 300's. Doubling a number in the 400's has a result in the 800's or 900's.)[If a is 1, c is 2 or 3. If a is 3, c is 6 or 7.]
When the digits are reversed, b replaces b so it must either be b * 2 = b or, if Rule 3 comes into play, b * 2 = 10 + (b - 1). [b could only be 0 when c < 5 and only 9 when c >4.]
With just four possibilities, there is no point in rendering additional rules! Just test them.
102? Is 204 = 201 + 1? No.
103? Is 206 = 301 + 1? No. [Wrong hundreds even.]
396? Is 792 = 693 + 1? No. [Wrong hundreds again.]
The answer given is the only one that passes.
As for extra points, an infinite number of 9's can be added between the 3 and the 7 without spoiling the major characteristics.
3999999997 * 2 = 7999999994 = 7999999993 + 1.
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Comments
Nice riddle, and a good explanation of the solution
This teaser was inspired by lessthanjake789 and his #33412.
Thanks.
Thanks.
Liked it a lot! Especially the extention (generalisation part).
I don't get the hint. Why would anyone say 123 * 2 = 321 + 1?
I figured it out much more quickly than using your equation (in about 10 seconds) and without any trial and error.
With a little thought, it was obvious that the middle digit had to be 9 and the ones digit had to be greater than 5 and 2 times the hundreds digit plus one. The one's digit of the last digit doubled needs to be one more than first digit. At this point I quit analyzing it because 3 and 7 were obvious.
The 9 is true because only zero and nine can double to become themselves. Zero won't work because then the first digit doubled would have to equal the last digit and the last digit doubled would have to equal the first digit plus 1, and this isn't possible with two integers.
Greater than 5 is true because it needs to carry a one when doubled.
Double the first digit plus one is true because the 9 will produce a carry when doubled.
Easy!
With a little thought, it was obvious that the middle digit had to be 9 and the ones digit had to be greater than 5 and 2 times the hundreds digit plus one. The one's digit of the last digit doubled needs to be one more than first digit. At this point I quit analyzing it because 3 and 7 were obvious.
The 9 is true because only zero and nine can double to become themselves. Zero won't work because then the first digit doubled would have to equal the last digit and the last digit doubled would have to equal the first digit plus 1, and this isn't possible with two integers.
Greater than 5 is true because it needs to carry a one when doubled.
Double the first digit plus one is true because the 9 will produce a carry when doubled.
Easy!
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