This puzzle has been shared with the claim that only geniuses can solve it.

11 × 11 = 4

22 × 22 = 16

33 × 33 = ?

It has gone viral on Facebook and the internet with millions of views as people have debated the correct answer.

Arguably there are many answers, as there are many patterns that fit the given information. However, there are 2 main answers that are most popular. I will go over what many people believe to be the correct answer, and I will explain how the 2 main approaches are flavors of the same idea.

Watch the video for an explanation.

**Can You Solve The Viral 11×11 = 4 Puzzle? The Correct Answer Explained**

Or keep reading.

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**Answer To Viral Puzzle 11×11 = 4**

Most people believe the correct answer is 36. The method to obtain the answer is to take the product of the sum of digits in each number being multiplied.

That is:

*aa* × *aa* → (*a* + *a*)(*a* + *a*)

This procedure matches the pattern of the puzzle:

11 × 11 → (1 + 1)(1 + 1) = 4

22 × 22 → (2 + 2)(2 + 2) = 16

And it suggests the answer of 36.

33 × 33 → (3 + 3)(3 + 3) = 36

This “product of the sum of digits” is what many people believe to be the correct answer. But there is debate.

**Alternate answer: 18**

Other people thought about the puzzle in terms of doing the multiplication and then taking the sum of the digits in the answer. In other words, this is finding the “sum of the digits in the product.”

11 × 11 = 121 → 1 + 2 + 1 = 4

22 × 22 = 484 → 4 + 8 + 4 = 16

This procedure suggests an answer of 18.

33 × 33 = 1089 → 1 + 0 + 8 + 9 = 18

The “sum of the digits of the product” gives 18, while the “product of the sum of the digits” gives 36.

It seems like these two methods are completely different. However, there is a way to see they are flavors of the same concept. And in this way, it is possible to get an answer of 36 when doing the “sum of the digits of the product.”

**Getting 36 from the sum of the product**

Let’s go deeper into the details of how to calculate the product of two numbers and how to sum the digits in the answer.

The number 11 can be written as 10 + 1, so we have:

11 × 11

= (10 + 1)(10 + 1)

= 1(100) + 2(10) + 1(1)

= 121

The digits in the answer are the coefficients of the sums of powers of 10, which is how decimal numbers are written. The sum of the digits in the answer is 1 + 2 + 1 = 4.

Similarly, the number 22 can be written as 20 + 2, so we have:

22 × 22

= (20 + 2)(20 + 2)

= 4(100) + 8(10) + 4(1)

= 484

Again, the sum of the digits is the sum of the coefficients of the terms attached to powers of 10. The sum is 4 + 8 + 4 = 16.

What happens with 33 then? The number 33 can be written as 30 + 3, so we have:

33 × 33

= (30 + 3)(30 + 3)

= 9(100) + 18(10) + 9(1)

What happens if you add up the terms attached to powers of 10? You get 9 + 18 + 9 = 36. You get the answer of 36 from this procedure!

But isn’t the answer supposed to be 18 from this method? Yes, the reason is 18(10) is larger than 100, so it involves carryover. We can simplify the answer as:

9(100) + 18(10) + 9(1)

= 9(100) + 10(10) + 8(10) + 9(1)

Now 10(10) = 100, so that contributes 1 more term to the 100 value.

9(100) + 10(10) + 8(10) + 9(1)

= 10(100) + 8(10) + 9(1)

Now 10(100) is equal to 1000, so we again have carryover.

10(100) + 8(10) + 9(1)

= 1(1000) + 0(100) + 8(10) + 9(1)

= 1089

This gives the familiar answer of 1089, which is what a calculator would show for 33 × 33.

But we can see that 9(100) + 18(10) + 9(1) is a valid representation of the product, and the sum would be 36 if we do not go through the carryover process.

So we have found a connection between the two methods.

*aa* × *aa* → (*a* + *a*)(*a* + *a*) = product of sum of digits = sum of the product (without carryover)

It is possible to justify the answer of 36 from either method.

**Other ways to arrive at 36**

In the video, I show the same thing visually using diagrams from the “multiply by lines” method. The answer in each case is the number of intersections of the lines or “dots” in the figure, and 33 × 33 has 36 dots.

Via MindYourDecisions YouTube

The key to the answer of 36 is the multiplicative nature of the procedure. Start with 11 × 11 = 4 as a given. The second line has two terms that are 2 times 11, so the answer should be 2(2) = 4 times as large. The third line has two terms that are 3 times 4, so the answer should be 3(3) = 9 times as large.

11 × 11 = 4

22 × 22 = (2 × 11)(2 × 11) = 4(11 × 11) = 4(4) = 16

33 × 33 = (3 × 11)(3 × 11) = 9(11×11) = 9(4) = 36

There is another method to illustrate the multiplicative property and avoid carryover: express the answer in terms of a specific modulus, such as 39.

11 × 11 = 4 mod 39

22 × 22 = 16 mod 39

33 × 33 = 36 mod 39

All of these methods suggest that 36 is the correct answer to the puzzle.

**Quora discussion**

https://www.quora.com/Puzzles-and-Trick-Questions-What-is-the-correct-answer-to-11×11-4-22×22-16-33×33

Read more:

Viral Puzzle 11×11 = 4. The Correct Answer Explained