Multiplying Two Digit Numbers Without Writing Anything Down

Four methods, the split method, squaring numbers ending in 5, multiplying by 11, and the difference of squares, cover almost any two-digit multiplication.

Multiplying two two-digit numbers without writing anything down is possible for almost any pair once you have three or four methods available and the judgment to pick the fastest one for the numbers in front of you. Straight repetition of the standard written algorithm in your head is slow because it asks working memory to hold too many partial sums at once. The methods below each remove a step by exploiting some structure in the specific numbers, whether that is a shared distance from a round number, a factor of 11, or an ending digit of 5, and knowing which one applies is what makes the calculation fast.

The most general method breaks one number into tens and units and distributes the other across both parts. Twenty three times 47 becomes 20 times 47 plus 3 times 47, which is 940 plus 141, giving 1,081. This works for any two-digit pair and is the fallback method when none of the faster tricks below apply, though it asks you to hold two three-digit numbers in memory before adding them.

Choosing which number to split changes how hard the addition at the end is. Splitting 47 instead of 23 in the same problem gives 23 times 40 plus 23 times 7, which is 920 plus 161, also 1,081, but the two partial products are closer in size and slightly easier to add. Splitting the number with the rounder tens digit usually produces the cleaner pair of partial products.

Any two-digit number ending in 5 squares using a two-step rule: multiply the tens digit by the next integer up, then attach 25 to the end. Sixty five squared takes the tens digit 6, multiplies it by 7 to get 42, and attaches 25, giving 4,225. The actual multiplication, 65 times 65, would otherwise require the full split method, so this shortcut turns a two-digit-by-two-digit multiplication into a single-digit multiplication plus a fixed ending.

The same rule holds at every scale. Thirty five squared is 3 times 4, which is 12, with 25 attached, giving 1,225, and eighty five squared is 8 times 9, which is 72, with 25 attached, giving 7,225. The rule works because a number ending in 5 can be written as 10 times its tens digit plus 5, and squaring that expression algebraically always produces exactly this pattern.