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Grex > Iq > #143: Math and Logic Puzzles |  |
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| Author |
Message |
| 15 new of 32 responses total. |
flem
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response 18 of 32:
| 
Feb 27 22:28 UTC 2000 |
resp:17 contains a solution.
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aruba
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response 19 of 32:
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Feb 28 14:58 UTC 2000 |
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 x would work? An extra
bonus would be given if the number x contains no more than 12 digits.
|
aruba
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response 20 of 32:
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Mar 2 02:01 UTC 2000 |
Well, all valid inputs end with 2. So if x produces the reverse of x, then
the reveres of x begins with a 2. So it might be the case that the last thing
that happens is to tack a 2 onto the output. I.e., x might begin with a 7.
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flem
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response 21 of 32:
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Mar 2 19:11 UTC 2000 |
This one was more difficult, but x = 745361745362 does the trick.
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aruba
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response 22 of 32:
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Mar 2 19:19 UTC 2000 |
Looks good. 734173412 also does it.
The Fourth Test:
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?
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flem
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response 23 of 32:
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Mar 3 01:19 UTC 2000 |
Ah. I had figured out the 5361 sequence before I got around to playing with
4's. I probably should have done the fours first, but...
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flem
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response 24 of 32:
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Mar 3 02:17 UTC 2000 |
This one is pretty easy. There's a 5 digit solution, which might not use the
techniques that the problem is designed for, but is correct.
I'm pretty sure that there's only one 5 digit solution, so this might
be a pretty big hint, but I'll not post it just yet.
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aruba
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response 25 of 32:
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Mar 3 03:03 UTC 2000 |
I got a 5-digit solution too. It's not the solution he gives in the book.
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flem
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response 26 of 32:
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Mar 3 18:43 UTC 2000 |
There's a class of solutions that are inputs of the form x31x312, where
x meets some conditions. The most obvious of these is where x = "", the
empty string: 31312 -> 3131. I'd be interested to see if the book
gives any solutions not of this form.
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aruba
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response 27 of 32:
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Mar 4 04:13 UTC 2000 |
The solution in the book points out that 5361x2 yields x1x, so 53615362 yields
5361536.
I believe I can prove that 31312 is the only 5 digit solution. (At least,
I think I proved it in the shower this morning, but I didn't write it down.)
The Fifth Test:
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?
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flem
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response 28 of 32:
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Mar 6 17:45 UTC 2000 |
The easy way (to prove that 31312 is the shortest input that satisfies
test 4) would be exhaustive testing. There aren't many valid inputs
that are 5 digits or fewer: x1yz2, where x is in { ,1,3,4,5,6,7},
and y and z are in { ,1,2,3,4,5,6,7}. (y and z might possibly be 8,9,
or 0; I don't recall if the rules allow that or not.) That's only
7*8*8 = 448 valid inputs of 5 digits or less. (847 if y and z can be
8, 9 or 0.)
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aruba
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response 29 of 32:
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Mar 6 19:54 UTC 2000 |
It's OK to use digits 8, 9, and 0.
Other valid 5-digit inputs look like xy1z2. But the only ones of those with
4-digit outputs are 331x2, which produce xxxx, and therefore don't qualify.
The only way to get a 4-digit output from x1yz2 is if x is 3, and then you
have 31yz2, which produces yzyz. So clearly 31312 is the only such which
produces itself minus its last digit. So there you go. :)
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aruba
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response 30 of 32:
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Mar 7 19:47 UTC 2000 |
Hint on the 5th test (in #27): One way to do this one is to look for an input
x whose output is 1x2.
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aruba
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response 31 of 32:
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Mar 11 18:32 UTC 2000 |
Here's another hint:
/---------------------------------------------\
| Find a prefix p such that px2 yields 1xx22. |
\---------------------------------------------/
|
aruba
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response 32 of 32:
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Mar 17 23:00 UTC 2000 |
We should try to wrap this up by the end of winter agora, so I'll solve test
number 5. If p = 6477431, then px2 yields 1xx22, which in turn yields xx2.
So therefore pp2 yields 1pp22, which yields pp2. So one solution is
/-----------------\
| 647743164774312 |
\-----------------/
Onward...
/-----------------------------------------------------\
| The Sixth Test: The suitor had to send a number x, |
| get back a number y, return y to the princess, and |
| get back the *reverse* of the original number x. |
| What number x would work? |
\-----------------------------------------------------/
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