The 7 th challenge of the Royal English Mathematical Society presented in the newspaper "El País" goes over the keys of a piano. Here you can see the problem statement, both video and statement.
is a piano in which they are pressing a few keys following a series of numbers. Do you press, then Re, and pulse skip a Fa, skip two, and so on.
The questions are very simple and refer to when we hold 7000 notes:
1. How many keys we will have played under a C note?
2. Are there any notes that have not been down in no time?
The answer must be send before from 12 pm today. I read on Thursday and, for ghosts, I would say it took 5 minutes to find the solution as far congruence and I found it easy to find a pattern.
I leave it on if you want to copy and send.
My proposed solution is:
We believe that there are 7 notes, which we number from 1 to 7 (OD = 1, RE = 2, MI = 3, FA = 4, G = 5, LA = 6, SI = 7). As a cycle of 7 notes, the key will match the 8 th 1 st, 9 th to the 2 nd and so on. We work with congruences in module 7.
T1 = 1 T2 = T1 +1 = 2
T2 T3 = T4
+2 = 4 +3 = 7 = T3 = T4 T5
+4 = 11 = 4 -> For the 11 th note coincides with the 4 th note
T6 = T5 +5 = 9 = 2 -> For the 9 th note coincides with the 2 nd note
T7 = T6 +6 = 8 = 1 -> For the 8 th note coincides with the 1 st note
T8 = T7 = T7 +7 +0 (Because 7 keys to scroll is not moving as any since it is the same grade) = 1
T9 = T8 = T8 +8 +1 = 2
...
is noted that a cycle of repeated notes in 7 keys. This is because it is congruent modulo 7 +4 to -3, -2 and +5 to +6 to -1.
note then that the notes would be DO, RE, FA, IF, FA, RE, DO, DO, RE, FA, SI, ...
Replies: 1.How
each cycle of 7 keys pressed 2 times the note of, in 7000 will pulse 2000 times. 2.The
notes never would strike would not MI, SOL, LA.
I hope you serve at least for curiosity's sake it is not normally a problem as mathematical puzzle.
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