Mimir Mental Math is an app for iPhone, iPod touch, iPad and Mac.
It not only contains a huge collection of mental math tricks, you can also practice each trick. And there are not only these tricks in the app, I have also included general strategies for math problems where none of the tricks can be used.

The app has two parts: the lessons and the training. In the lessons you can read all the texts on how to do the math tricks and then practice them. There are 27 lessons at the moment (with up to 3 tricks per lesson), for:
-Addition
-Subtraction
-Multiplication
-Division
-Divisibility
-Squaring
-Calculate the weekday for any date

In the second part of this app, the training, you can solve random math problems with increasing difficulty. The goal here is not only to become better and faster at solving math problems, you will also learn to detect which trick would be the best for the current math problem. This is why there is a help button in the training-section. If you push this button, the application will search for the trick or tip that is possibly the easiest for solving the current math problem.

Additional features:
-Two languages: All texts are in english and german.
-User management: Multiple users can use the same device. The application saves the settings and position in the training for each user separately.
-Two different GUI-designs: Use a cool blue flickering display with white text, that looks like from a sci-fi movie. Or switch to a simple white paper background with black text.

Mental Math Tricks

Here are some of the texts contained in the app. It is just a small portion of the tricks that you will learn in the application.

Multiplication

Single digit multiplication

Multiplication of a one digit number with another one digit number is relatively simple. But it is also the basis of all the more complicated multiplications.

After some training, you will have memorized every combination. Therefore you do not really need tricks to do these multiplications, but there still are a few simple methods for some of the numbers.

Multiplication by 2:

When multiplying smaller numbers by 2, simply add the number to itself.

Multiplication by 4 or 8:

Instead of multiplying a number by 4, you can also multiply it twice by 2. Or three times by 2, if you want to multiply by 8.

Multiplication by 5:

Multiplying an even number by 5 always ends with 0, and multiplying an odd number by 5 always ends with a 5. So simply look at the other number.
If it is even, divide it by 2 and add a 0 to the right of it. If it is odd, subtract 1 from it, divide it by 2 and add a 5 to the right.

Example:

7 x 5 =

Step 1: 7 is odd: 7 - 1 = 6
Step 2: 6 ÷ 2 = 3
Step 3: 7 is odd: add a 5 to the end

Result: 35

Multiplication by 9:

It is easy to memorize all possible single digit multiplications by 9, because there is a simple pattern in these results. The tens-digit of the result is always one less than the original number. And if you add the ones-digit to the tens-digit, it always equals 9.

Example:

7 x 9 =

Step 1: Tens-digit: 7 - 1 = 6
Step 2: Ones-digit: 9 - 6 = 3

Result: 63 If you have a complete blackout when you need to multiply a single digit number by 9, you can do the following:

Put both your hands in front of you with the fingers stretched. Let's say you want to calculate 6 x 9. Count to the 6th finger and fold it down. Now you simply need to count the number of fingers that are still standing on the left side of the 6th finger, and also the ones on the right side, and you will know the result.

In the case of 6 x 9, you will have 5 fingers on the left, and 4 fingers on the right, so the result is: 54.

Multiplication by 5

To multiply a number by 5, it is sometimes easier to multiply it by 10 and then divide it by 2.

Because a multiplication by 10 is very easy (just move every digit to the left and attach a 0 if there are no decimals), and division by 2 is not very difficult either, this can be easier and faster than directly multiplying with 5.

Example:

389 x 5 =

Step 1: 389 x 10 = 3890
Step 2: 3890 ÷ 2 = 1945

Result: 1945

This also works when multiplying larger numbers by 5. Multiplying an even number by 5 always ends with 0, and multiplying an odd number by 5 always ends with a 5.
So multiplications by 5 can also be solved like this:
Look at the number that has to be multiplied by 5. If it is even, divide it by 2 and add a 0 to the right of it. If it is odd, subtract 1 from it, divide it by 2 and add a 5 to the right.

Example:

389 x 5 =

Step 1: 389 is odd: 389 - 1 = 388
Step 2: 388 ÷ 2 = 194
Step 3: 389 is odd: add 5 to the right of the result

Result: 1945

Multiplication by 10

To multiply a number by 10 is very easy. Just add a 0 to the right of the number. If there are decimals, simply move the decimal point to the right.

Example:

849 x 10 =

Step 1: Add 0 to the right of the number: 8490

Result: 8490

Multiplication by 11

Multiplications by 11 are great to show friends how good you are at math, because these multiplications look far more complicated than they are.

To multiply 11 with numbers below 10, simply use the same number as the tens-digit. For example, 3 x 11 is 33, 6 x 11 is 66 etc.

For larger numbers, the following method can be used:

Let us start with a easy example: 43 x 11
In this case, just move the 4 to the left, sum the two numbers (4 + 3) and write the resulting number between the two others:
4 7 3

So 473 is the result of 43 x 11. Easy, right?

It was easier in this case because the sum of 4 and 3 is not greater than 9, so there is nothing to carry over. If the digits do sum to 10 or higher, simply carry over the 1 (as the maximum sum here is 9 + 9 = 18, the carry-over will never be greater than 1) and add it to the digit to the left.

Example:

85 x 11 =

Step 1: 8 + 5 = 13
Step 2: 8 13 5
Step 3: Carry 10 from 13 over to the 8:
935

Result: 935

This method also works for larger numbers.

Example:

5983 x 11 =

Step 1: Move the digits on the side outwards:
5___3

Step 2: Add each digit to the digit to its right:
5
5 + 9 = 14
9 + 8 = 17
8 + 3 = 11
3

Step 3: Add the carry-overs to the digit on the left side:
5 + 1 = 6
4 + 1 = 5
7 + 1 = 8
1
3

Result: 65813

The rightmost digit always stays the same, so doing this from right to left is actually a bit easier.
For longer numbers, going from left to right is more useful (because you have to memorize less), but you have to be more careful with the carry-overs.

Example:

99999 x 11 =

Step 1: 9 9+9 9+9 9+9 9+9 9
Step 2: Carry-overs: 9+1 8+1 8+1 8+1 8 9

Result: 1099989

Multiplying numbers between 11 and 19

There is a special method for multiplying two numbers that are both between 11 and 19.

Let's use the example:

13 x 17 =

Step 1: Add the ones-digit of the second number to the complete first number (or vice versa): 13 + 7 = 20

Step 2: Multiply the result of step 1 by 10: 20 x 10 = 200

Step 3: Multiply the ones-digits of both numbers: 3 x 7 = 21

Step 4: Add the results of step 2 and 3 to get the final result:

200 + 21 = 221

Another example:

15 x 19 =

Step 1: 15 + 9 = 24
Step 2: 24 x 10 = 240
Step 3: 5 x 9 = 45
Step 4: 240 + 45 = 285

Result: 285

Multiplying numbers close to a power of 10

This method works for any multiplication, but it only really makes sense if both numbers are close to a power of 10 (numbers close to 10, 100, 1000, etc.). It works like this:

Example:

93 x 86 =

Step 1: Use that power of 10 which is nearest to both numbers as the base (in this case: 100)

Step 2: Calculate for both numbers the difference between the base and the number. You can use the method of complements here to do this faster.
100 - 93 = 7
100 - 86 = 14

Step 3: Subtract one of the differences found in step 2 from the other original number (it doesn't matter which one).
93 - 14 = 79 (or 86 - 7 = 79)
This will give you the left part of the result (or in other words, multiply this number by the base for the first part):
79 x 100 = 7900

Step 4: For the right part of the result, multiply the two differences found in step 2:
7 x 14 = 98

Step 5: Put these two numbers together to form the result: 79 98 (or 7900 + 98)

Result: 7998

If the result of step 4 is greater than the base, remember to carry over.

Example:

5 x 6 =

Step 1: Base: 10
Step 2:
10 - 5 = 5
10 - 6 = 4
Step 3: 6 - 5 = 1 (or 5 - 4 = 1)
Step 4: 5 x 4 = 20
Step 5: 1 as the left part, and 20 as the right part, so carry over the 2 from the right to the left part. This of course is not necessary if you multiply the left part by the base and then add up both numbers (1 x 10 + 20).
Result: 30

If both numbers are above the base, you have to add the difference in step 3 instead of subtracting it.

Example:

112 x 105 =

Step 1: Base: 100
Step 2:
112 - 100 = 12
105 - 100 = 5
Step 3: 112 + 5 = 117 (or 105 + 12 = 117)
Step 4: 5 x 12 = 60
Step 5: 117 60

Result: 11760

If only one of the numbers is above the base and the other one is below it, you have to either add the difference of the greater number in step 2, or subtract the difference of the lower number.

And the right part found in step 4 has to be subtracted from the left part multiplied by the base.

Example:

125 x 98 =

Step 1: Base: 100
Step 2:
125 - 100 = +25
98 - 100 = -2
Step 3: 125 - 2 = 123 (or 98 + 25 = 123)
Step 4: 2 x 25 = 50
Step 5: 123 x 100 - 50 =

Result: 12250 A good way here to remember when to add and when to subtract is to remember that minus multiplied by minus results in a positive number, the same as when multiplying a positive number with a positive number.

But if you multiply a negative number with a positive number (as in the last case, when one number is below the base and the other above it), you get a negative result, so you need to subtract in step 5.

Division

Division by 5

A division by 5 can be simplified by multiplying the number by 2 and then dividing by 10.

Example:

3165 ÷ 5 =

Step 1: 3165 x 2 = 6330
Step 2: 6330 ÷ 10 = 633

Result: 633

Division by 9

You can use remainders for faster division by 9.

If you want to find the remainder of a division by 9, simply sum all the digits of the number. For example, if you divide 6824 by 9, you get 6 + 8 + 2 + 4 = 20 = 2 + 0 = 2, so 2 is the remainder.

This works because for each tens, divided by 9, there is a remainder of 1. And for each hundreds, there is also a remainder of 1 (because the next lower divisible number is 99), and also for each thousands etc...

So, by summing all the digits, you are actually summing all the remainders.

This knowledge can be used for divisions by 9. Let's use 62 ÷ 9 as a simple example. First, take the leftmost digit (6) and use it for the result. Then calculate the sum of the remainders: 6 + 2 = 8.
So 62 ÷ 9 equals 6 with a remainder of 8. Easy, right?

If the sum of the remainder is greater than 9, divide the remainder by 9 and carry this result over to the last digit of the final result. Use the remainder of the last division for the next sum.

Example:

837 ÷ 9 =

Step 1: Use the first number for the result: 8
Step 2: 8 + 3 = 11, which is greater than 9, so 11 ÷ 9 = 1, remainder 2
Step 3: Carry over the 1 and add it to the first number: 92
Step 4: 2 + 7 = 9, so carry over 1: 93, remainder 0

Result: 93

An example with remainder:

3443 ÷ 9 =

Step 1: First number: 3
Step 2: 3 + 4 = 7 ? 37
Step 3: 7 + 4 = 11; 11 ÷ 9 = 1, remainder 2 -> 382
Step 4: 2 + 3 = 5

Result: 382, with a remainder of 5

Division by 10

If there is a 0 on the right side of the number, simply remove it to divide by 10. If not, add a decimal point to the number between the last digit and the rest (or, if there already is a decimal point, move it to the left).

Example:

93481 ÷ 10 =

Result: 9348.1

Divisibility

Divisibility by 2

A number is divisible by 2 if the last digit of the number is even (0, 2, 4, 6 or 8).

Example:

46236758

Check: is the last number even?

Divisibility by 3

A number is divisible by 3 if the sum of its digits is divisible by 3. For larger numbers, continue adding the digits of the sum until a single-digit number is reached. If that single-digit number is 0, 3, 6 or 9, then the original number is divisible by 3.

Example:

46236759

sum of all digits:
4 + 6 + 2 + 3 + 6 + 7 + 5 + 9 = 42

sum again:
4 + 2 = 6

Check: Is the final sum 0, 3, 6 or 9?

Divisibility by 4

To know if a number is divisible by 4, you only need to check if the number formed by the last two digits is divisible by 4. Because 100 is divisible by 4, you can simply ignore everything except these two last digits.
With these two digits, if the first digit (the tens-digit) is even, the other digit must be 0, 4 or 8. And if the first digit is odd, then the last digit must be 2 or 6 to be divisible by 4.

Example:

46236756

Only the last two digits:
56

Check: The first digit (5) is odd, so is the last digit 2 or 6?

Divisibility by 5

A number is divisible by 5 if the last digit is a 0 or a 5. Easy, isn't it?

Example:

46236755

Check: Is the last digit a 5 or a 0?

Squaring

Square numbers ending in 5

To square a number ending in 5, do the following:

Step 1: Take the number formed by the other digits (except the 5).
Step 2: Multiply that number by itself + 1.
Step 3: Attach a 25 to the left of the result.

Example:

85²

Step 1: 8
Step 2: 8 x (8 + 1) = 72
Step 3: 72 25

Result: 7225

Square numbers between 50 and 59

There is a simple way to square numbers between 50 to 59:

Step 1: Add the ones-digit to 25.
Step 2: Square the ones-digit.
Step 3: Attach the result of step 2 to the right of the result of step 1 (as 2 digits, so if the result was 9, attach 09)

Example:

52²

Step 1: 25 + 2 = 27
Step 2: 2² = 4
Step 3: 2704

Result: 2704

Square numbers between 26 and 75

There is a special method to square numbers between 26 and 75, but you need to know (or have a different way to calculate) all the squares of the numbers 1 to 25 in order for it to be useful.

Step 1: Take the difference between the number and 25, multiplied by 100.
Step 2: Square the difference between the number and 50.
Step 3: Add the results of step 1 and step 2.

Example:

61²

Step 1: 61 - 25 = 36
36 x 100 = 3600
Step 2: (61 - 50)² = 121
Step 3: 3600 + 121 = 3721

Result: 3721

Square numbers close to 1000

To square numbers close to 1000, there is a similar method to the one for numbers close to 100:

Step 1: Take the difference between the number and 1000.
Step 2: If the number is larger than 1000, add the result of step 1 to the number. Otherwise subtract it.
Step 3: Multiply the result of step 2 by 1000.
Step 4: Square the difference and add it to the result of step 3.

Example:

986²

Step 1: Difference between 1000 and 986: 14
Step 2: 986 is smaller than 1000: 986 - 14 = 972
Step 3: 972 x 1000 = 972000
Step 4: 14² + 972000 = 196 + 972000 = 972196

Result: 972196