EPT"My daughter is bery choosy," said the sultan, "and wantsto marry only someone who shows extraordinary intelligence. So if you want to marry her, you must first pass eight tests."
"What are the tests?" asked the suitor.
"Well, for the first test, you have to write down a number that will be sent to the princess. She will then send back a number to you. If she sends back the very same number that you have sent her, then she will allow you to take the second test. But if her number is different from yours, then you are out."
"Now, how can I possibly know what number to write?" asked the suitor. "How can I guess what number the princess has in mind?"
"Oh, she doesn't have a certain number in mind," said the sultan. "The number she sends back is dependent on the number you send. The number you send completely determines the number she will send back. And if you send the right number, then she will send back the same number."
"Then how can I guess the right number?" asked the suitor.
"It's not a matter of guessing," said the sultan. "You must deduce the correct number from the rules I am about to give you. For any numbers x and y, by xy I mean not x times y but x followed by y, both numbers, of course, written in base ten arabic notation. For example, if x is 5079 and y is 863, then by xy I mean 5079863. Now here are the rules:
Rule 1: For any number x, if you write her 1x2 then she will send you back the number x. For example, if you write 13542, she will write back 354.
Rule 2: For any number x, the repeat of x means xx. For example, the repeat of 692 is 692692. And now, the second rule is that is x brings back y, the 3x will bring back the repeat of y. For example, since 15432 brings back 543, then 315432 will bring back 543543. From which it further follows that if you send her 3315432, you will get 543543543543.
Rule 3: The reverse of a number means the number written backwards. For example, the reverse of 62985 is 58926. The third rule is that if x brings back y, then 4x brings back the reverse of y. For example, since 172962 brings back 7296, the number 4172962 brings back 6927.
Now if we combine the first three rules...
since 316982 brings back 698698, then 4316982 brings back 896896, the reverse of what 316982 brings back.
Rule 4: (The erasure rule) If x brings back y, and if y contains at east two digits, then 5x brings back y with the first digit erased. For example, since 13472 brings back 347, 513472 brings back 47.
Rule 5: (The addition rule(s)) If x brings back y, then 6x brings back 1y and 7x brings back 2y. For example, since 15832 brings back 583, then 615832 brings back 1583 and 715832 brings back 2583.
"Those are the rules," said the sultan, "and from them can be deduced a number x that will bring back the very number x. There are actually an infinite number of solutions, but any single one will suffice for passing the first test."
"Are there any meanings to these numbers?" asked the suitor.
"Ah, that is the princess' secret, but fortunately you don't have to know the meaning in order to pass the first test."
So, can you figure out the number that the prince would have to send that would return itself?
The rest of the tests are as follows, but I don't suggest trying them until you figure out the one before it.
Test 2: For the second test, the suitor had to send the princess a number, x, such that she would send back the repeat of x, (the number xx). What number would work?
Test 3: For the third test, the suitor had to send the princess a number x such that she would send back the reverse of x. What number would work? An extra bonus would be given if the number x contains no more than twelve digits. What number would work?
Test 4: For this test, the suitor had to send a number x such that the princess would send back the number x with its last digit erased. What x would work?
Here is where it gets tricky!
Test 5: For this test, the suitor had to send a number x such that the princess would send back a different number y, which the suitor was to send back to the princess, and she would (hopefully) send back the first number x. What number x would work?
Test 6: The suitor now had to send a number x, get back a number y, return y to the princess, ad get back the reverse of the original number x. What number x would work?
Test 7: The suitor now had to send a number x, get back a number y, return y to the princess, and get back the number x with the first and last digits swapped. What number x would work?
Test 8: For the final test, the suitor was to send a number x, the princess would then send back a number y, the suitor was then to send back the reverse of y, the princess would then send back a number in the form zz (a number z repeated), the suitor was then to bread zz in half (so to speak) and send her back z. The princess would then (hopefully) send back the original number x. What number would work?
