Digital Problems are puzzles where the solution consists only of digits, i.e. 0 through 9. That means the search space will be constrained initially to [0..9] for each variable.  
The Torn Number 
from "Amusements in Mathematics, Dudeney", number 113
I had the other day in my possession a label bearing the number 3025 in large figures. This got accidentally torn in half, so that 30 was on one piece and 25 on the other. On looking at these pieces I began to make a calculation, scarcely concious of what I was doing, when I discovered this little peculiarity. If we add the 30 and the 25 together and square the sum we get as the result the complete original number on the label! Now, the puzzle is to find another number, composed of four figures, all different, which may be divided in the middle and produce the same result. 

The General Store 
from "Mathematical Puzzles of Sam Loyd", number 30
The owner of a general store, who is something of a puzzlist, has put up this sign to see if any of his mathematical friends can translate it properly. Each different letter stands for a different digit. The words above the horizontal line represent numbers that add to the total of "ALL WOOL". The problem is to change all the letters to the correct digits. C H E S S + C A S H + B O W W O W + C H O P S + A L S O P S + P A L E A L E + C O O L + B A S S + H O P S + A L E S + H O E S + A P P L E S + C O W S + C H E E S E + C H S O A P + S H E E P _______________ A L L W O O L 

Safe Cracking  from the Oz Primer, page 17
The code of Professor Smart's safe is a sequence of 9 distinct nonzero digits C1 .. C9 such that the following equations and inequations are satisfied: C4  C6 = C7 C1 * C2 * C3 = C8 + C9 C2 + C3 + C6 < C8 C9 < C8and C1 < > 1, C2 < > 2, ..., C9 < > 9 

Twin Letters 
In the following puzzle, there are ten pairs of letters to be assigned
to the same digit so that the multiplication (including intermediate results)
is correct. Can you find out the pairs and their values?
A B C * D E F ____________ G H I J K L M N O ____________ P Q R S T 

Send more money etc.  (various sources)
These puzzles are the classic "cryptarithmetic" problems. Find digits to replace the letters, so that each letter stands for a different digit and the substitution yields a correct addition. S E N D + M O R E _________ M O N E Y ...and in german: N A S E N + S A L B E ___________ B R E N N Tand
W E I N + W E I B _________ L I E B Ea bit more complex is the twodimensional variation ABC + AA = DEF : + : GF : C = A = = = FH + DF = IID 