A man asks his father about his age.

Instead of giving a direct answer the father wrote down two positive integer numbers on a piece of paper and says:

- Sum these two numbers and evaluate the square root of the total. Doing this, youŽll get my age.

The son takes his pocket calculator and inadvertently types the two numbers, one after the other, without pressing the "+" button (e.g. if the numbers were 124 and 357, he types 124357, instead of the sum of the numbers).

After this he makes a second mistake by pressing the square root button, not once, but twice.

Finding an integer number as the result he shows it to his father:

- This is your age.

- No, youŽre wrong, but the number you found is precisely the age of your mother, who is older than me.

How old is the father?

Note: While a solution is trivial with the aid of a spreadsheet, can you derive it without one?

The following program raises possible ages of the mother to the fourth power. It then divides the resulting number into two parts, and adds the two parts. If the result is a perfect square and its square root is less than the assumed mother's age, the results are printed.

DEFDBL A-Z
CLS
FOR mother = 20 TO 120
  m4 = mother * mother * mother * mother
  m$ = LTRIM$(STR$(m4))
  FOR i = 1 TO LEN(m$) - 1
    a = VAL(LEFT$(m$, i)): b = VAL(MID$(m$, i + 1))
    fsq = a + b
    f = INT(SQR(fsq) + .5)
    IF f * f = fsq THEN
      IF f < mother AND f > 19 THEN
       PRINT f; mother, LEFT$(m$, i); " "; MID$(m$, i + 1)
      END IF
    END IF
  NEXT i
NEXT mother

The results follow:

 F   M        a    b
25  50        625 0000
36  60       1296 0000
56  67       2015 1121
49  70       2401 0000
64  80       4096 0000
81  90       6561 0000

where a and b are the numbers the father asked to be added.

However, in all but one instance the father would have had to specify that four zeros be used to enter the zero, which is outside the bounds of what's expected.

So the father is 56, the mother is 67 and the father asked the son to add 2015 and 1121.

Posted by Charlie on 2008-08-04 16:33:44