A set of seven positive integer numbers has the following properties:
The mode of the set is 1.
The mean of the set is 4.
The median of the set is 5, and occurs just once.
The range of the set is 6.
What are the numbers?
(In reply to
Puzzle Answer by K Sengupta)
The median of the set is 5, occurring only once. Thus the set is:
* * * 5 * * *
The range of the set is 6 and the mode of the set is 1.
Thus, the maximum integer is 1+6=7
Since the digit 5 occurs only once, we need to have at least two 1s for the mode to equal 1.
Therefore, the set is
1 1 * 5 * * 7
This directly implies that:
(5th position, 6th position) = (6, 6), or (6,7)
In the former case, the given set admits of precisely two 6s
In the latter case, the given set admits of exactly two 7s.
Therefore, for the mode to equal 1, we need one more 1 (in the 3rd place)
So the set is either:
1 1 1 5 6 6 7, or:
1 1 1 5 6 7 7
The former case gives a total of 27, so that the mean is 27/7 which is NOT equal to 4. This leads to a contradiction.
The latter case gives a total of 28, so that the mean is 28/7 which is equal to 4.
Consequently, the required set of 7 positive integers is given by:
{1, 1, 1, 5, 6, 7, 7}
Edited on August 21, 2022, 2:43 am
Explanation to Puzzle Answer
