2019/finals/strings_as_a_service.html

<p>
Carlos has been working in technology so long that he's starting to feel a bit
burnt out. Hoping to rejuvenate himself, Carlos has been seeking out more
artistic opportunities.
</p>

<p>
Yamaha, the well-known creator of musical apparatus, has approached Carlos with 
a request that might be right up his alley: they'd like him to design a brand new
instrument. Immediately, Carlos knows what to do.
</p>

<p><em>
"You may have seen Pat Metheny's 42-string guitar, but that's nothing compared
to what we're going to make together."
</em></p>

<p>
Carlos presents his plan for a 1,000-string guitar, complete with programmatic
tuning so that you don't need to turn 1,000 knobs by hand. Yamaha's market
research suggests that these sorts of guitars would be great for playing
palindromic chords, chords where the first string plays the same note as the
last string, the second string plays the same note as the second-to-last
string, and so on. Carlos is quickly tasked with developing default tunings for
the strings so that the guitars are ready to play right out of the box.
</p>

<p>
For various integers <strong>K</strong>, Carlos wants to find a set of at most 1,000 strings on which
exactly <strong>K</strong> distinct palindromic chords can be played. The guitar's strings are
arranged in a line, and each one must be tuned to a note from the set {A, B, C, D, E, F, G}.
A chord is then played by strumming a contiguous subset of
1 or more strings. Two chords are considered to be distinct if there is at least one 
string that is used in one chord but not the other; chords involving the same notes but
different strings are considered different.
</p>

<p>
For example, if <strong>K</strong> = 9, a set of 7 strings could be tuned to the notes
C, A, B, B, A, G, E in order from left to right. You can play 7 different palindromic
chords by strumming single strings, the chord BB by strumming the 3rd and 4th
strings, and the chord ABBA by strumming the 2nd, 3rd, 4th, and 5th strings.
This is a total of 9 distinct palindromic chords.
</strong>

<p>
Output any non-empty string of valid musical notes, with length at most 1,000, 
representing the tunings of sequential strings. An aspiring musician must be able
to play exactly <strong>K</strong> distinct palindromic chords on these strings. It's guaranteed 
that there is at least one valid output for each possible valid input.
</p>


<h3>Input</h3>

<p>
Input begins with an integer <strong>T</strong>, the number of tunings that Carlos needs to figure out.
<br />For each tuning, there is a single line containing the integer <strong>K</strong>.
</p>


<h3>Output</h3>

<p>
For the <em>i</em>th tuning, print a line containing "Case #<em>i</em>: " followed by a string of up to 1,000 characters representing
a tuning of strings as described above on which exactly <strong>K</strong> distinct palindromic chords can be played.
</p>


<h3>Constraints</h3>

<p>
1 &le; <strong>T</strong> &le; 500 <br />
1 &le; <strong>K</strong> &le; 100,000 <br />
</p>

<h3>Explanation of Sample</h3>

<p>
In the first case, "ACE" is a valid output as it contains exactly 3 palindromes: "A", "C", and "E". On the other hand, "DAD" would not be valid as it contains 4 palindromes.
</p>

<p>
In the second case, "GAGA" is a valid output as it contains exactly 6 palindromes: "G", "A", "G", "A", "GAG", and "AGA".
</p>

<p>
<strong><i>Note that other outputs would also be accepted for each sample case.</i></strong>
</p>