### 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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