Find the pattern in 1x1=1, 11x11=121, 111x111=12321

Question

Observe the pattern:

$1 \times 1 = 1$
$11 \times 11 = 121$
$111 \times 111 = 12321$

Find the pattern, explain why it works, and predict: what is $1111 \times 1111$ ?

Solution — Step by Step

Identify the pattern in the results

The results are: 1, 121, 12321.

Write them out digit by digit:

$1 \times 1$ : digits are 1
$11 \times 11$ : digits are 1, 2, 1
$111 \times 111$ : digits are 1, 2, 3, 2, 1

The pattern is clear: the digits rise from 1 up to the number of 1s in the multiplier, then fall back to 1 symmetrically. It's a "mountain" pattern of digits.

So $1111 \times 1111$ should be 1234321 (digits 1, 2, 3, 4, 3, 2, 1).

Verify the prediction

Let's verify: $1111 \times 1111$ .

$1111 \times 1111 = 1111 \times (1000 + 100 + 10 + 1)$

$= 1111000 + 111100 + 11110 + 1111$

$= 1234321$ ✓

The pattern holds.

Understand WHY this works — using algebra

Write the number with $n$ ones as a sum:

$\underbrace{11\ldots1}_{n} = \sum_{k=0}^{n-1} 10^k = \frac{10^n - 1}{9}$

So $\underbrace{11\ldots1}_{n}^2 = \left(\frac{10^n - 1}{9}\right)^2 = \frac{(10^n - 1)^2}{81}$

$(10^n - 1)^2 = 10^{2n} - 2 \times 10^n + 1$

So $\underbrace{11\ldots1}_{n}^2 = \frac{10^{2n} - 2 \times 10^n + 1}{81}$

This is elegant but requires long division to see the digit pattern. Let's use a more direct approach.

Understand via carrying — the real reason

When we multiply 111 × 111, we're summing three shifted copies of 111:

    111
  × 111
  -----
    111       (111 × 1, units place)
   1110      (111 × 10, tens place)
  11100     (111 × 100, hundreds place)
  -----
  12321

When we add these column by column:

Units column: one 1 → sum = 1
Tens column: two 1s → sum = 2
Hundreds column: three 1s → sum = 3
Thousands column: two 1s → sum = 2
Ten-thousands column: one 1 → sum = 1

The columns get contributions from 1, 2, 3, 2, 1 ones — giving exactly the mountain pattern 12321.

For $n$ ones, each column gets contributions from 1 through $n$ and back to 1 (as long as no column sum exceeds 9, so no carrying occurs).

When does the pattern break?

The pattern fails when $n \geq 10$ , because a column sum would reach 10 or more, causing a carry that disrupts the clean digit pattern.

$\underbrace{11\ldots1}_{9}^2 = 12345678987654321$ (perfect!)

$\underbrace{11\ldots1}_{10}^2 = ?$ The middle column has ten 1s summing to 10, causing a carry → the pattern breaks.

Understanding when patterns break is as important as recognising them.

Why This Works

The visual "mountain" pattern arises from the geometry of long multiplication: multiplying a number by $\underbrace{11\ldots1}_{n}$ creates $n$ shifted copies of the number, and the overlap structure of these copies creates the triangular count of 1s in each column.

This kind of digit pattern is a favourite in olympiad-style questions and Class 6 NCERT exercises because it trains students to look for structure rather than just compute.

Alternative Method

Think of it recursively: $\underbrace{11\ldots1}_{n} = \underbrace{11\ldots1}_{n-1} \times 10 + 1$ .

So $\underbrace{11\ldots1}_{n}^2 = (\underbrace{11\ldots1}_{n-1} \times 10 + 1)^2 = \underbrace{11\ldots1}_{n-1}^2 \times 100 + 2 \times \underbrace{11\ldots1}_{n-1} \times 10 + 1$ .

Starting from $1^2 = 1$ , each step builds the next result from the previous one. This is a recursive derivation.

Common Mistake

⚠️ Common Mistake

Students often memorise the pattern without understanding the carry rule. If someone asks $\underbrace{11\ldots1}_{11}^2$ , they might (incorrectly) predict digits going up to 11 then back down. But once a column sum reaches 10, carrying happens and the pattern changes. Never extend a pattern beyond its valid range without checking.

💡 Expert Tip

This question tests pattern recognition — a key skill in maths olympiads and competitive exams. Always: (1) state the pattern, (2) verify with the next term, (3) explain WHY it works. A three-part answer always scores better than just stating "the next number is 1234321."

Question

Solution — Step by Step

Why This Works

Alternative Method

Common Mistake

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