P Q, and R are all different double digit positive integers. P Q, and R are all less than 24. The mean of P Q, and R is prime. The median of P Q, and R is square. The range of P Q, and R is square. Find the values of P Q, and R .

The values of P,Q and R have to be between 10 and 23 as they are all contain two digits and are less than 24. The median of P,Q and R is a square number and the only square number in the 10-23 range is 16. So one of P,Q or R is 16. For the sake of simplicity we can say that Q is 16. The mean of the numbers should also contain double digits as the mean is a measure of the average in a data set and the lowest number is 10. Furthermore the mean is a prime number so the mean can be 11,13,17,19, or 23. The range of P, Q and R is a square number. The square numbers could be 1, 4 or 9 as anything above that is impossible to make with the above numbers. 11 13 17 19 23Multiplying by 3 gives us the sum of all three numbers33 39 51 57 69 Since on of the numbers is 16 the sum of the sum of the other two numbers ( P and R) are:7 23 35 41 53.P and R have to be above and below the median (16 = Q) as P Q and R are different. The range cannot be 1 as P and R have to be above and below 16 and have to have a difference of 4 or 9. The numbers that fit this criteria to give a range of 4 are 17 and 13, 18 and 14, 19 and 15 . The numbers that fit this criteria to give a range of 9 are 21 and 12, 22 and 13 and 14 and 23. The only two numbers to make the sum of P and R are 22 and 13. So P, Q and R are 13, 16 and 22