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
thenamust be 0 orbmust be 1. It is easy to find out which case applies. We look at other lines that includeaandb. For example, ifa=0thenc + a = cand ifb=1thenc + b = d. Similarly, froma ÷ b = afollowsa=0orb=1and from each ofcd × b = cd,cd ÷ b = cdfollowsb=1. 
a + b = cd
then it follows thatc=1because the sum of two 1digit 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 fromcd − 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, care all different, so none of them can be 1 or 0. Thus, one ofa, bmust be 2 and the other one 3 or 4 andcis 6 or 8. The same conclusion can be drawn fromc ÷ b = a.

a × a = b
thenbis a square number unequalaandb<10, soa=2, b=4ora=3, b=9. 
c + ea = eg
then the first digit (the tens) ineaand inegare the same, so we conclude thatc + a = g. 
c + ea = fg
then the first digit (the tens) ineaand infgare different. This can only be due to a carry over, so we conclude thate + 1 = fandc + a = 10 + g.
Try to find more clues, for example:

What can be concluded from
a × a = ba? Which values cana, bonly have?

What follows from
eb × c = cd? Which value caneonly have?

If you know that g = 1then what doesfg ÷ c = dtell you aboutf, c, d?
Have fun!
Video solutions with more hints are available for the following calcrostics problems from Caribou contests: