Colorful Cat's Play
Felicia the cat was playing with a pink yarn ball on the room floor of
square tiles; each row of tiles was colored in the rainbow spectrum: red, orange, yellow, green, blue, indigo, & violet respectively.
While the naughty cat was rolling the ball across the floor, the yarn thread was spun out, leaving its trail on the tiles, where Felicia started from the upper left red corner all the way down to the lower left violet corner in column 1.
To continue the play, Felicia would make an upward track alternating with a downward track in the next right columns, without receding to the left ever. For example, as shown in the picture, after the first move in column 1, it had to move to the next violet tile before making a new upward track. Any up or down-ward track in any column had to cover at least 2 colored tiles, so there was no going horizontally from one column to the next two.
Throughout the play, Felicia crossed each color of the tiles more than once and crossed 2 tiles in 2 non-adjacent columns while the other 5 tracks differed in lengths of tiles. In the end, it finished playing at the lower right violet corner.
If there were only one row that Felicia crossed all 7 tiles of the same color with no groups of over 8 connected uncrossed tiles, which color would that row be?
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1 solution
If we designate the colored rows with numbers 1-7 for red to violet respectively, the first move in the first column leaps the number 1 (red) to 7 (violet); it's a +6 direction.
From the scenario, Felicia moved for 2 tiles twice, so that would mean it equals to a magnitue of 1 tile direction, and the other different tracks would have 2, 3, 4, 5 tiles direction (the maximum is 6 in magnitude because there are 7 tiles in 1 column, and 6 is used in the first track).
Since the tracks end at the right violet corner again, the final number will be 7 (violet). All we need to do is to fill in the gaps with 1, 1, 2, 3, 4, 5:
1 + 6 - _ + _ - _ + _ - _ + _ = 7
Clearly, that means we have to arrange these six digits with three plus signs and three minus signs so that the result is 0 with 3 conditions: all colors are visited more than once; 1 & 1 can't be next to each other (The problem indicates that the moves are not adjacent.) and only one colored row all crossed by the cat.
Even though there may be quite a number of combinations, only two sets fit all the criteria:
1 + 6 - 4 + 1 - 3 + 2 - 1 + 5 = 7 (all yellow crossed only)
And 1 + 6 - 3 + 1 - 4 + 5 - 1 + 2 = 7 (all blue crossed only)
Among these two possibilities, only the latter one has all groups of uncrossed tiles under 8, so this is the solution.
Thus, Felicia crossed all the blue tiles in the yarn ball play.