Multiplying by eleven is easy. Here is the model: 11 X AB = A(A + B)B. For example 11 X 23 = 253 and 11 X 51 = 561. Sometimes there is a carry, as in 11 X 56 = 5(11)6 = 616. You can extend this to three or more digits with the model 11 X ABC = A(A + B)(B + C)C, so 11 X 123 = 1353. This technique can also be extended to double numbers as with 3 X 55 = (3 X 5) X 11 = 15 X 11 = 165. One final example: 55 X 77 = 11 X 11 X 35 = 11 X 385 = 3(11)(13)5 = 4235.
Squaring numbers that end in 5 is easy. Here is the model: (A5)^2 =A(A + 1)25. This means that 25^2 = (2 X 3)25 = 625, and 75^2 = (7 X 8)25 = 5625. Also notice that this extends to (A5 + N)(A5 - N) = A5^2 - N^2, so 24 X 26 = 25^2 - 1 = 624 and 33 X 37 = 1225 - 4 = 1221. This algorithm also works where the first digits are the same and the second digits add to 10. The model is AB X AC = A(A + 1)(B X C) if B + C = 10. This means that 23 X 27 = (2 X 3)(3 X 7) = 621.
If two numbers are both slightly larger than 100, the are easy to multiply. The model is 10A X 10B = 1(A + B)(A X B). This means that 102 X 103 = 1(2 + 3)(2 X 3) = 10506 and 107 X 109 = 16316. If the two numbers are both slightly less than 100, let A* mean 100 - A, then A X B = (A - B*)(A* X B*), So 93 X 97 = (93 - 3)(7 X 3) = 9021.
For squaring numbers slightly larger than 50, the model is 5A^2 =(25 + A)A^2. Therefore 57^2 = (25 + 7)7^2 = 3249. For numbers slightly less than 50, the model is (50 - B)^2 = (25 - B)B^2, so 47^2 = (25 - 3)3^2 = 2209. The same tricks work for these squares so 56 X 58 = 57^2 - 1 = 3249 - 1 = 3248.