Save the damsel!
One fateful night, Brenda was walking home from her friend's birthday party when she suddenly got ambushed by her greedy aunt! Of course why Brenda's aunt is so greedy for money that she kidnapped her own niece is another story. Brenda was pale with shock but thankfully still had her faculties intact.
Locking Brenda up behind a gate, Brenda's aunt barked, "I have abducted you! The only way you can escape is to crack the 10-digit code of the padlock. To help you, here are some facts:
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Every digit from 0 to 9 appears exactly once in the 10-digit code.
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The leftmost number is the sum of the two numbers on its immediate right.
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The leftmost number and the rightmost number differ by 1.
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Interestingly enough, the second left number and the second right number differ by 2, to establish a pattern.
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And more interesting still, just as the third left number and the third right number differ by 3, the fourth left number and the fourth right number differ by 4.
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The number 0 occupies one of the middle two positions.
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The sum of the left five numbers is a perfect square.
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The sum of the right five numbers is a prime number.
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The rightmost number is a perfect cube.
I believe I have given you enough information. Can you crack the code?
The answer is 7342096158.
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It is easiest to look at facts 7 and 8 first. The sum of all ten digits is known, that is 45. Split them up into two numbers such that one of them is a perfect square and our combinations are 1+44, 4+41, 9+36, 16+29, 25+20. The only combinations which consist of a perfect square and a prime number are 4+41 and 16+29. But 4+41 isn't practical as the five smallest numbers already add up to 0+1+2+3+4=10 which is way more than 4. Hence the only possibility is 16+29 (i.e. the five left digits add up to 16, the five right digits add up to 29).
Considering fact 9, the rightmost digit is 0, 1 or 8. But 0 is out as it contradicts with fact 6. If the rightmost digit is 1, then considering fact 3, the leftmost digit is either 0 or 2. As mentioned, 0 can be ruled out on the basis of fact 6. But if the leftmost digit is 2, then in consideration of fact 2, we cannot have a pair of numbers that add up to 2 without repetition (i.e. 2=2+0 or 1+1 both of which involve repetition of digits, thus violating fact 1).
Since the rightmost digit being 0 or 1 would lead to a contradiction, it must be 8. From fact 3, the leftmost digit is either 7 or 9. If the leftmost digit is 9, then considering fact 2, the two digits on its immediate right add up to 9 too but this means that the three left digits already sum to 18 which already exceeds the sum of the five left digits of 16. This means the leftmost digit must be 7. The second and third left digits add up to 7 too which gives us a total of 14 for the three left digits. We have 7????????8.
Hence the sum of the fourth and fifth left digits is 16-14=2. The only possible combination is 2+0. In consideration of fact 6, the fourth left digit is 2 while the fifth left digit is 0. From fact 5, we also deduce that since the fourth left digit is 2, the fourth right digit must be 2+4=6. We now have 7??20?6??8.
Next we deduce what the second and third left digits are. We see 7=3+4 or 2+5 or 1+6 but 2+5 is out as 2 is already in the fourth left position. Similarly, 1+6 is out too as 6 is already in the fourth right position. So we can only consider 3+4. But is it the 3 first or the 4 first? If the 4 is first (i.e. in the second left position), then considering fact 4, the second right digit is either 4-2=2 or 4+2=6 but 2 and 6 are already repeated. Hence the 3 must take the second left position and the 4 the third. Since 4 occupies the third left position considering fact 5 the third right digit must be 4-3=1. Similarly with 3 in the second left position the second right digit must be 3+2=5. We have 73420?6158.
Now look at the final question mark among the string of numbers as it should become clear to you that the question mark should be replaced by the number 9. Our answer is 7 3 4 2 0 9 6 1 5 8 .