13 purple, 15 yellow, and 17 maroon chameleons are found on an island. When two different-coloured chameleons come together, they both turn into the third colour.
Do all chameleons eventually have the same hue after a certain number of pairwise meetings?
Check if you were right - full answer with solution below. Ā
Solution:
ā¢ We are given 13 purple, 15 yellow, and 17 maroon chameleons. When two chameleons of different colors meet, both change into the third color.
ā¢ Consider the quantity (Yellow ā Purple). Initially:
Yellow ā Purple = 15 ā 13 = 2.
ā¢ After any meeting, this quantity either remains unchanged or changes by a multiple of 3. Hence, throughout the process:
Yellow ā Purple = 2 + 3k (for some integer k),
which means it is never divisible by 3.
ā¢ Now consider the final situation where all chameleons become the same color:
If all are purple: Yellow ā Purple = 0 ā 45 = ā45
If all are yellow: Yellow ā Purple = 45
If all are maroon: Yellow ā Purple = 0In all cases, the value is divisible by 3.
ā¢ This creates a contradiction because the value (Yellow ā Purple), which is never divisible by 3 during the process, must become divisible by 3 in the final state.
Therefore, it is not possible for all chameleons to eventually have the same color.
