Question 1048851: Professor A. M. Nesia has a safe with a combination lock. In her journal, the note she uses to help her remember is this diagram →
and the year of her birth, 1941,
This reminds her that the code is a sequence of five perfect squares (square numbers) in ascending order where the mean = 19, median = 4 and mode = 1.
Find the combination.
Answer by solver91311(24713)   (Show Source):
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Apparently, the arrow indicates ascending order, the 1941 is the mean, median, and mode, and she just remembers that it is a sequence of 5 perfect squares.
Since there is a definite mode, the number 1 must occur in the combination at least twice. Since 4 is the median, it is the third of the 5 numbers, and since there is room for only two smaller perfect squares, they must both be 1. So far we have 1 1 4 x y.
Since the mean of the five numbers is 19, the sum of the five numbers must be 5 times 19 or 95. Since we have already accounted for 1 + 1 + 4 = 6 of that total, the sum of x and y must be 89.
Now all you have to do is consider the series of perfect squares greater than 4 and less than 80 (because the 4th one must be at least 9) and find two of them that add up to 89.
9, 16, 25, 36, 49, 64
None of them end in zero, so 9 and 49 are eliminated as candidates. None of them end in 3, so 16 and 36 are eliminated as candidates.
John
My calculator said it, I believe it, that settles it
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