Zetamac is a popular online website as a benchmark for mental math.
It consists of solving as many addition, subtraction, multiplication, and division problems within 120s.
My goal score is to reach a score of 80, granted I'm quite the ways off so I'll be documenting my progression and new found strategies as I go.
I wanted to keep track of my progress so I created this website which stores and graphs my progress. Check it out: https://datamac.vercel.app/.
First things first, ensure you can answer anything within the 12x12 times table in 1s. Also remember, getting quicker comes from a lot of intuition that you build up through practice.
This is the easiest type. Typically summing digits left to right is always faster than right to left. It allows you to start typing the answer immediately, and you only need to worry about off by 1 fluctuations due to carry over.
For example, 38 + 55, I know it should start with an 8, then I see the ones digits sum to 13, so I quickly change my answer to 93.
This is a bit trickier than addition but quite similar. If it is a 2 digit subtract 2 digit, I will use the strategy as from addition, working right to left and being clever about carry overs. Another strategy is seeing how much you need to add to the smaller number to get to the larger value.
For example, 83 - 69. I can see I just need to add 14 to turn 69 into 83, so 14 is my answer.
If it's a 3 digit subtract 2 digit, I will determine what I need to add to that 2 digit to make it 100, then determine from 100 how much I need to add to get that 3 digit number.
For example, 132 - 79. I know I need 21 to make 79 become 100. Then I add 21 + 32 = 53 to get my answer.
Most often a single digit multiplied by the second number is solvable by rounding the second number to a nice value, then subtracting / adding as needed.
For example, 9x74. Here it is actually easier to round the single digit to 10 then subtract. So 740 - 74 = 666. Or breaking the problem to 9x70 + 9x4 = 666 is another strategy, sometimes faster since addition is less brain intensive that subtraction.
Now the challenging aspect comes when both digits are two digits.
If the first number is 11, add the digits in the second number, and insert that value in the middle of those two numbers.
For example, 11x63. 6+3=9, so the answer is 693. Now consider, 11x84. 8+4=12, so now we must do a carry over and insert the 2 in the middle. So it becomes 924.
Why does this work?
Under the hood, to solve 11x63, it is doing 10x63 + 63. We know 10x63 will always result in the number ending in 0. Then adding 63 on will guarantee the last digit stays the same. The middle digit is calculated by adding 6 + 3 which is why the middle digit is the sum of digits.
If the first number is 12, I typically break it down into 10 x number2 + 2 x number2.
For example, 12x45 = 10 x 45 + 2 x 45 = 540.
When dividing by 11 the answer is either the two outer digits or because of the carry over, the two outer digits but subtract the leftmost digit by 1.
For example, 374 / 11 = 34. This is simple case without carry over.
Now consider, 974 / 11 = 84. Here whenever we see the hundreds digit greater than or equal to the tens digit, subtract it by 1 since there was a carry over. So since 9 > 7, our answer is 84 instead of 94.
When dividing a 3 digit number by 9, there is a clever trick which I will show by example.
Consider 342/9. The answer is comprised of two digits, the first is floor(34/9), the second is 10-2. So the answer is 38. In other words, the first digit is how many times 9 goes into the two most significant digits, and the second digit is 10 - least significant digit.
I want to preface again these strategies work because Zetamac ensures results are always integers, so these rules aren't generalizable to any values.
Another important strategy, for the smaller number, memorize the sequence of 10xnumber, 20xnumber, etc and also the multiples of a number.
For example, 540/12. I know instantly that 480 = 40 x 12. Now I need an additional 60 which is 12x5. So the answer is 45.