The video explains a mathematical trick involving three-digit numbers. It starts by demonstrating the process with an example: pick a three-digit number where the first and last digits are different (e.g., 731). Reverse the number (137), subtract the smaller from the larger (731 - 137 = 594). Then, reverse the result (495) and add it to the previous result (594 + 495 = 1089). The video then shows that this process consistently yields 1089, even with a different starting number (e.g., 532). The explanation delves into the algebra behind this phenomenon. A three-digit number is represented as 100a + 10b + c, and its reverse as 100c + 10b + a. Subtracting these results in 99*(a-c), indicating that the middle digit 'b' cancels out and the result is always a multiple of 99. A "housekeeping rule" is mentioned: if the difference between the outer digits is 1, the subtraction yields 99, which should be written as 099 to maintain three digits. The video highlights that all three-digit multiples of 99 share a characteristic: the middle digit is 9, and the sum of the outer digits is also 9 (e.g., 594, where 5+4=9). This property holds even for 099. Therefore, after the subtraction step, the number will always be in the form p9q, where p + q = 9. When this number is reversed and added, the sum is consistently 1089 because the structure p9q, with p+q=9, leads to (100p + 90 + q) + (100q + 90 + p) = 101p + 180 + 101q = 101(p+q) + 180. Since p+q=9, this becomes 101*9 + 180 = 909 + 180 = 1089. The core takeaway is that the specific digits don't matter as much as the relationship between the outer digits, which is constrained to sum to nine after the subtraction step.

BriTheMathGuy

Pick any three-digit number. Just make the first and last digits different. Let's use 731. Reverse it and subtract the smaller from the larger. That makes 594 in this case.

Now, reverse that and add. You get 1089. And you probably think this worked because 731 works. Well, let's pick a different number. How about 532? Reverse and subtract, then reverse and add. It's 1089 again.

Every starting number you could choose lands on those same four digits. So, what's going on here? Well, if you write your number as 100a + 10b + c, its reverse is 100c + 10b + a. How about subtracting these? You get 99 * a - c, and that middle digit is gone. The whole subtraction only cares about the gap between the outer two digits. So, the result is always 99 * something.

There is a housekeeping rule here. Keep the three digits. If your outer digit differs by just one, the subtraction hands you 99. Write it as 099 and keep going. And every three-digit multiple of 99 has the same fingerprint, a nine in the middle and the outer digits summing to nine. Look at 594. The middle's a nine, and 5 + 4 = 9. Even 099 plays by this rule.

So, no matter what number you started with, the number you're about to reverse and add looks like p9q, where p + q is nine. Writing it out, you get this, and p + q is locked at nine, so that's 101 * 9 + 180, which is 1089. The actual digits never really mattered, only their sum, and their sum can't be anything but nine.

The 1089 Trick

Pick any three-digit number. Just make
the first and last digits different.
Let's use 731. Reverse it and subtract
the smaller from the larger.
That makes 594 in this case. Now,
reverse that and add. You get 1089. And
you probably think this worked because
731 works. Well, let's pick a different
number. How about 532? Reverse and
subtract, then reverse and add. It's
1089 again.
Every starting number you could choose
lands on those same four digits. So,
what's going on here?
Well, if you write your number as 100a +
10b + c, its reverse is 100c + 10b + a.
How about subtracting these?
You get 99 * a - c, and that middle
digit is gone.
The whole subtraction only cares about
the gap between the outer two digits.
So, the result is always 99 * something.
There is a housekeeping rule here. Keep
the three digits. If your outer digit
differs by just one, the subtraction
hands you 99.
Write it as 099 and keep going.
And every three-digit multiple of 99 has
the same fingerprint, a nine in the
middle and the outer digits summing to
nine.
Look at 594.
The middle's a nine, and 5 + 4 = 9. Even
099 plays by this rule.
So, no matter what number you started
with, the number you're about to reverse
and add looks like p9q,
where p + q is nine.
Writing it out, you get this, and p + q
is locked at nine, so that's 101 * 9 +
180, which is 1089.
The actual digits never really mattered,
only their sum, and their sum can't be
anything but nine.

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The 1089 Trick

Transcription