CD To Tape
Matilda wanted to transfer her favourite CD to tape.
The CD has six songs and the length of the tracks are 7:55, 9:40, 9:15, 12:45, 8:20 and 11:30; a total playing time of 59:25. After changing the order of the songs, Matilda was able to fit all the songs, without any breaks, on a sixty minute tape. How did she arrange the songs?
For the songs to fit on the tape, each side must be almost completely filled, as the total playing time will leave 35 seconds unused on a sixty minute tape. If the two longest songs were placed on the same side, 12:45 and 11:30 take up 24:15 together, leaving only 5:45, which is not enough for any other single song. So they must go on opposite sides.
By trial we get:
Side 1 12:45, 7:55 and 9:15 (29:55 total)
Side 2 11:30, 9:40 and 8:20 (29:30 total)
If the total playing time is less than the length of the tape, is a solution always possible?