Detective Work on Calcrostic Problems

Solving a calcrostic is like checking out a crime scene for clues. The following are examples, how to get clues from single lines of the puzzle.

For example, if the row, column or diagonal is

•  a × b = a
  then a must be 0 or b must be 1. It is easy to find out which case applies. We look at other lines that include a and b. For example, if a=0 then  c + a = c and if b=1 then  c + b = d. Similarly, from  a ÷ b = a follows a=0 or b=1 and from each of  cd × b = cd,  cd ÷ b = cd follows b=1.
•  a + b = cd
  then it follows that c=1 because the sum of two 1-digit numbers can not be more than 9+9=18 and if both are different then not more than 9+8=17. The same conclusion can be made from  cd − b = a.
  
•  a × b = c
  then what is special is that the result is only a one digit number, so less than 10. Also, a, b, c are all different, so none of them can be 1 or 0. Thus, one of a, b must be 2 and the other one 3 or 4 and c is 6 or 8. The same conclusion can be drawn from  c ÷ b = a.
  
•  a × a = b
  then b is a square number unequal a and b<10, so a=2, b=4 or a=3, b=9.
•  c + ea = eg
  then the first digit (the tens) in ea and in eg are the same, so we conclude that  c + a = g.
•  c + ea = fg
  then the first digit (the tens) in ea and in fg are different. This can only be due to a carry over, so we conclude that  e + 1 = f and  c + a = 10 + g.

Try to find more clues, for example:

• What can be concluded from  a × a = ba ? Which values can a, b only have?
  
• What follows from  eb × c = cd ? Which value can e only have?
  
• If you know that g = 1 then what does  fg ÷ c = d tell you about f, c, d ?

Have fun!

Video solutions are available for the following calcrostics problems from Caribou contests:
Nov 2011, grade 7/8 question 23
Oct 2011, grade 7/8 question 17
May 2011, grade 7/8 question 17