Behold the solution to Doctor E. Strikes Back #6. If you have forgotten the question, click here to see it again.
(0) |
The number contains exactly 2 different digit values. |
| (1) | No 2 digits differ by 4, 5, or 6. |
| (2) | More than 1 digit is a square. |
| (3) | Exactly 1 digit value appears more than once. |
| (4) | All digits are even. |
| (5) | The number is a palindrome. |
| (6) | No prime digit divides a different-valued digit. |
| (7) | No digit value appears more than twice. |
| (8) | The number contains all digit values between the largest and smallest. |
| (9) | All adjacent digits differ by 1. |
There were several ways to get to the unique answer 567898765. (The number 987656789 does not work, because more than 1 digit is a square in this number. If you read the wording of the problem carefully, there is a distinction between "digits" and "digit values").
(0) |
FALSE: There are more than two digit values |
| (1) | FALSE: The first and the fifth digit differ by 4 |
| (2) | FALSE: The only square is the fifth digit |
| (3) | FALSE: Several digit values appear more than once |
| (4) | FALSE: The first digit, for example, is odd |
| (5) | TRUE: 567898765 backwards is 567898765 |
| (6) | TRUE |
| (7) | TRUE |
| (8) | TRUE: All digit values from 5 through 9 appear |
| (9) | TRUE |
The best explanation for how to get the answer comes to us courtesy of Jimmy B:
"I started by assuming (0) was true: The number contains exactly 2 different digit values. Therefore only one other number is true and the rest are false which leads to contradictions in each case. Thus (0) is false. Assuming (9) is true, (8) must also be true because (9) alone won't work, and since (0) is false (7) is also true. Now (4) is false, therefore (0) through (3) are also false because (8) is true. Since (1) is false, (5) must be true then (6) is also true." Jimmy doesn't tell us how he got the ordering, but once you know that the number is a palindrome (5 true) with the 9 only appearing once (2 False), the statements guide you to 567898765.
Eric P. decided to just write a computer program that started at 1, and kept checking numbers until it found a solution. Click here to see his program.
William Tucker was born March 19th, 1619.
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