Visual Patterns - Free Printable

Let’s solve each pattern one by one. We’ll look at how the number of items grows from step to step, find a rule (formula), and then use that rule to check if the given total for a later step is correct — or figure out what step it corresponds to.

We’re told the final counts (like 3,784 segments, 85 blocks, etc.) and we need to verify them or understand how they were reached. But since the problem doesn’t ask a specific question, I’ll assume the task is: For each pattern, find the formula for the nth term, and confirm that the given number matches some step n.

Let’s go pattern by pattern.

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PATTERN 1: Squares made of segments

Step 1: 1 square → 4 segments
Step 2: 2x2 grid → 12 segments? Wait, let’s count carefully.

Actually, in a grid:

- Step 1: 1x1 square → horizontal lines: 2 rows × 1 segment = 2; vertical lines: 2 columns × 1 segment = 2 → total 4
- Step 2: 2x2 grid → horizontal: 3 rows × 2 segments = 6; vertical: 3 columns × 2 segments = 6 → total 12
- Step 3: 3x3 grid → horizontal: 4 rows × 3 = 12; vertical: 4 columns × 3 = 12 → total 24

Wait — but the image shows Step 1: 1 square (4 segments), Step 2: 2x2 (which has 12 segments?), Step 3: 3x3 (which should be 24?).

But the label says “3,784 segments”. Let’s find the formula.

In an n x n grid of squares:

- Number of horizontal segments: (n+1) rows × n segments each = n(n+1)
- Number of vertical segments: same = n(n+1)
- Total = 2n(n+1)

Check:
- n=1: 2×1×2 = 4 ✔️
- n=2: 2×2×3 = 12 ✔️
- n=3: 2×3×4 = 24 ✔️

So formula: S(n) = 2n(n+1)

Set equal to 3784:

2n(n+1) = 3784
→ n(n+1) = 1892

Solve n² + n - 1892 = 0

Discriminant: 1 + 4×1892 = 1 + 7568 = 7569
√7569 = 87 (since 87² = 7569)

n = [-1 + 87]/2 = 86/2 = 43

✔ So at step 43, there are 3,784 segments. Correct.

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PATTERN 2: Green blocks (L-shapes)

Step 1: 1 block
Step 2: 3 blocks (looks like L with 2 on bottom, 1 on top right)
Step 3: 6 blocks? Wait, let’s see:

Actually, looking at the shapes:

Step 1: 1 block
Step 2: adds 2 more → total 3
Step 3: adds 3 more → total 6
Step 4: adds 4 more → total 10

This looks like triangular numbers: T(n) = n(n+1)/2

Check:
- n=1: 1
- n=2: 3
- n=3: 6
- n=4: 10

Yes! So formula: B(n) = n(n+1)/2

Given: 85 blocks

Set n(n+1)/2 = 85
→ n(n+1) = 170

Try n=13: 13×14=182 → too big
n=12: 12×13=156
n=13 is 182, so no integer solution? Wait, maybe I miscounted.

Wait — look again at the image:

Pattern 2:

First shape: 1 cube
Second: 3 cubes (like a corner)
Third: 6 cubes? Actually, third looks like 1+2+3 = 6?
Fourth: 1+2+3+4 = 10? Yes.

So yes, triangular numbers.

But 85 is not a triangular number? Let’s check:

T(13) = 13×14/2 = 91
T(12) = 78
T(13)=91, T(12)=78 → 85 is between them. That can’t be.

Wait — maybe it’s different.

Alternative: Maybe it’s cumulative sum of odd numbers? No.

Or perhaps the pattern is: each step adds a row and column?

Another idea: Look at the 3D view.

Step 1: 1
Step 2: 1 + 2 = 3
Step 3: 3 + 3 = 6
Step 4: 6 + 4 = 10 → still triangular.

But 85 is not triangular. Unless... wait, maybe the labeling is off.

Perhaps the first figure is step 0? Try:

If step n starts at n=0:

n=0: 0? No, first is 1.

Maybe it’s n(n+1)/2 for n starting at 1, and 85 is wrong? But the problem says “85 blocks” — perhaps it’s for step 13? But 13×14/2=91.

Wait — let me recalculate: Is 85 a triangular number?

Solve n(n+1)/2 = 85 → n² + n - 170 = 0

Discriminant: 1 + 680 = 681 → √681 ≈ 26.1 → not integer.

Hmm. Problem.

Wait — perhaps I misidentified the pattern.

Look again: The second pattern shows:

- First: single cube
- Second: three cubes in L shape (two on base, one stacked on end)
- Third: six cubes? Actually, it looks like a staircase: 1 + 2 + 3 = 6
- Fourth: 1+2+3+4=10

Yes.

But 85 is not triangular. Unless... maybe it's the sum of first n odds? 1+3+5+... = n². 85 is not square.

Another thought: Perhaps the "blocks" include hidden ones? Unlikely.

Wait — maybe the pattern is different. Let’s count the visible blocks in the images:

Image for Pattern 2:

Leftmost: 1 block
Next: 3 blocks (arranged as two side by side, one on top of right one)
Next: 6 blocks? It looks like a 3-step staircase: bottom row 3, middle 2, top 1? No, in the image it’s shown as increasing L-shape.

Actually, upon closer inspection (from standard problems), this is often the "corner" or "gnomon" pattern where each step adds a new layer.

Standard sequence: 1, 3, 6, 10,... which is triangular.

But 85 is not triangular. Unless the given number is for a different step.

Perhaps the formula is n(n+1)/2, and we set it to 85, but since it’s not integer, maybe the problem has a typo? Or I need to reexamine.

Wait — another possibility: Maybe the first figure is step 1 with 1 block, step 2 with 4 blocks? No, image shows 3.

Let’s calculate what n gives close to 85.

T(12) = 78
T(13) = 91

85 - 78 = 7, not matching.

Perhaps it’s not triangular. Let’s think differently.

Notice that in 3D, sometimes these are pyramids.

Another idea: The number of blocks might be the sum of the first n natural numbers, which is triangular, but perhaps for n=13, it’s 91, and 85 is a mistake? But the problem states "85 blocks", so likely it’s correct for some n.

Wait — let’s try n=13: 13*14/2=91
n=12:78
Difference is 13, so no.

Perhaps the pattern is: each step adds n blocks, but starting from n=1.

Same thing.

I recall that in some patterns, the L-shape might be counted differently.

Let’s assume the formula is T(n) = n(n+1)/2, and 85 is approximately T(13)=91, but that’s not helpful.

Wait — perhaps the "85 blocks" is for step 13, and I miscalculated the pattern.

Let’s look online or recall: In many textbooks, this exact pattern (green blocks growing L-shape) is triangular numbers, and for example, step 10 is 55, step 13 is 91.

But 85 is given. 85 = 5*17, not helpful.

Another thought: Maybe it’s the number of unit cubes in a solid that is n layers high, with each layer being a rectangle.

For example, step 1: 1x1 =1
Step 2: 2x2 minus something? No.

Perhaps it’s the sum of the first n odd numbers, but that’s n².

85 is not square.

Let’s move on and come back.

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PATTERN 3: Blue squares (staircase)

Step 1: 3 squares? Image shows: first has 3 squares (2 on bottom, 1 on top left) — actually, it’s a 2x2 minus one? No.

Looking:

First figure: 3 squares (like a small L or stairs)
Second: 6 squares? Let’s count:

Actually, standard staircase pattern:

Step 1: 1+2 = 3 squares
Step 2: 1+2+3 = 6 squares
Step 3: 1+2+3+4 = 10 squares

So same as triangular numbers but shifted.

Formula: S(n) = (n+1)(n+2)/2 ? For n=1: 2*3/2=3, n=2:3*4/2=6, n=3:4*5/2=10, yes.

So S(n) = (n+1)(n+2)/2

Given: 990 squares

Set (n+1)(n+2)/2 = 990
→ (n+1)(n+2) = 1980

Let k = n+1, then k(k+1) = 1980

k² + k - 1980 = 0

Discriminant: 1 + 7920 = 7921
√7921 = 89 (since 89²=7921)

k = [-1 + 89]/2 = 88/2 = 44

So n+1=44, n=43

Check: S(43) = 44*45/2 = 990 ✔️

Correct.

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PATTERN 4: Orange X-shapes

Step 1: 5 squares (center and four arms)
Step 2: 9 squares? Image: larger X, seems like 5 + 4 = 9?
Step 3: 13 squares? 9 + 4 = 13

So arithmetic sequence: 5, 9, 13,... common difference 4

Formula: A(n) = 5 + (n-1)*4 = 4n +1

Check:
n=1: 4*1+1=5 ✔️
n=2: 9 ✔️
n=3: 13 ✔️

Given: 173 squares

Set 4n +1 = 173
→ 4n = 172
→ n = 43

✔ Correct.

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PATTERN 5: Cyan circles (growing shape)

Step 1: 4 circles (square)
Step 2: 9 circles? Image: looks like 3x3 minus corners? Or plus extensions.

Count:

First: 4 circles
Second: 9 circles? Actually, it looks like a diamond or something.

From the image:

Step 1: 4 circles in a square
Step 2: 9 circles — perhaps 3x3 grid? But arranged in a cross or something.

Actually, standard pattern: this might be centered square numbers or something else.

Notice:

Step 1: 4
Step 2: 9
Step 3: 16? Image shows more.

Third figure has many circles. Let’s estimate.

Perhaps it’s (n+1)^2

n=1: 4 = 2^2
n=2: 9 = 3^2
n=3: 16 = 4^2? But image shows more than 16.

Look: third figure has a central part and arms.

Another idea: Each step adds a ring.

Step 1: 4
Step 2: 4 + 5 = 9?
Step 3: 9 + 7 = 16? Not matching.

Perhaps it’s the number of circles in a hexagonal packing or something.

Let’s think differently. From the image, the shape is symmetric and grows outward.

I recall that in some patterns, this is the "centered octagonal" or other, but let’s calculate the given number: 3,787 circles.

Assume a formula.

Suppose it’s quadratic: C(n) = an² + bn + c

From step 1: n=1, C=4
Step 2: n=2, C=9
Step 3: n=3, C=? Let’s count from image.

In the third figure for Pattern 5, it looks like a large circle cluster. Approximately, if step 1 is 2x2, step 2 is 3x3, step 3 is 4x4, then C(n) = (n+1)^2

Then for n=1:4, n=2:9, n=3:16, etc.

But 3787 is not a perfect square. √3787 ≈ 61.54, 61^2=3721, 62^2=3844, not 3787.

So not square.

Perhaps it’s the sum of first n odd numbers starting from 3 or something.

Another idea: The shape might be a diamond with diagonals.

For example, in a diamond shape, number of points is 2n(n+1) +1 or something.

Let’s search for a pattern.

Notice that in step 1: 4 circles
Step 2: 9 circles — difference 5
Step 3: let’s say 16, difference 7, so differences increase by 2, so quadratic.

Assume C(n) = an² + bn + c

n=1: a + b + c = 4
n=2: 4a +2b +c =9
n=3: 9a+3b+c = ?

From image, step 3 has how many? Looking closely, it seems like a 5x5 grid minus corners or something, but let's assume from growth.

If differences are 5, then next difference 7, so step 3: 9+7=16

Then:

Eq1: a+b+c=4
Eq2: 4a+2b+c=9
Eq3: 9a+3b+c=16

Subtract Eq1 from Eq2: (4a+2b+c) - (a+b+c) =9-4 → 3a+b=5 ...(i)

Eq3-Eq2: (9a+3b+c)-(4a+2b+c)=16-9 → 5a+b=7 ...(ii)

(ii)-(i): (5a+b)-(3a+b)=7-5 → 2a=2 → a=1

From (i): 3(1)+b=5 → b=2

From Eq1: 1+2+c=4 → c=1

So C(n) = n² + 2n +1 = (n+1)^2

But earlier we saw 3787 is not a square. Contradiction.

Unless the step numbering is different.

Perhaps step 1 is n=0.

Try n=0: C=4? But usually n starts at 1.

Another possibility: The first figure is step 0 with 1 circle? But image shows 4.

Let’s look at the image again mentally.

In Pattern 5, the first figure is 4 circles in a square. Second is 9 circles — likely 3x3. Third is probably 4x4=16, but the image shows a shape that is not full square; it has arms.

Actually, upon second thought, in many such problems, this pattern is the "number of circles in a plus sign" or "cross" that grows.

For example:

Step 1: center and 4 directions, but only 1 in each direction? So 1 (center) + 4*1 =5? But image shows 4, no center.

Image shows for step 1: 4 circles, no center, just the arms.

Perhaps it's 4n for n=1, but 4, then 8, not 9.

I think I need to accept that for step n, it might be (2n)^2 or something.

Let’s calculate what n would give 3787 if C(n) = (n+1)^2, but 61.54^2=3787, not integer.

Perhaps it's n(2n+2) or other.

Another idea: The shape is a square with side n+1, but with additional circles on the sides.

Let’s give up and look at the number 3787.

Factor 3787: divide by small primes.

3787 ÷ 7 = 541, since 7*541=3787? 7*500=3500, 7*41=287, total 3787, yes.

541 is prime? Probably.

Not helpful.

Perhaps the formula is different.

Let’s consider that in step n, the number of circles is 4n^2 or something.

For n=1: 4*1=4
n=2: 4*4=16, but image shows 9, not 16.

No.

Another approach: Perhaps the pattern is the same as Pattern 1 but for circles, but unlikely.

Let’s skip and come back.

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PATTERN 6: Toothpicks forming triangles

Step 1: 1 triangle → 3 toothpicks
Step 2: 4 small triangles (large triangle divided into 4) → how many toothpicks?

In a large triangle divided into 4 small ones (side 2), number of toothpicks:

Horizontal: 3 rows: top row 1, middle 2, bottom 3? Standard count.

For a triangular grid with side n (number of small triangles per side), number of toothpicks.

For n=1: 3
n=2: 9 toothpicks? Let's see: the large triangle has 3 sides, each of length 2, but shared.

Standard formula: for a triangular matchstick arrangement with side n, number of matchsticks is (3n(n+1))/2

Check:
n=1: 3*1*2/2=3 ✔️
n=2: 3*2*3/2=9 ✔️
n=3: 3*3*4/2=18

Image shows step 3: large triangle divided into 9 small ones, which should have 18 toothpicks.

Given: 2,838 toothpicks

Set 3n(n+1)/2 = 2838
→ n(n+1) = 2838 * 2 / 3 = 5676 / 3 = 1892

Same as Pattern 1!

n(n+1) = 1892

As before, n=43, since 43*44=1892

✔ Correct.

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PATTERN 7: Trees

Step 1: 3 trees
Step 2: 6 trees? Image: first has 3, second has 6 (3 on bottom, 3 on top? Or 2+4? Let's see.

Actually, from image:

Step 1: 3 trees in a row
Step 2: 6 trees — perhaps 3 on bottom, 3 on top offset
Step 3: 9 trees? Or more.

Looks like each step adds 3 trees.

So arithmetic sequence: 3, 6, 9, ... so T(n) = 3n

Given: 87 trees

3n = 87 → n=29

✔ Correct.

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PATTERN 8: Penguins

Step 1: 2 penguins
Step 2: 5 penguins? Image: first has 2, second has 5 (2 on bottom, 3 on top? Or 1+4)

Count:

Step 1: 2
Step 2: 5
Step 3: 9? Image shows more.

Differences: 5-2=3, if next is 9, difference 4, so perhaps quadratic.

Assume P(n) = an² + bn + c

n=1: a+b+c=2
n=2: 4a+2b+c=5
n=3: 9a+3b+c=?

From image, step 3 has 9 penguins? Let's assume.

Then:

Eq1: a+b+c=2
Eq2: 4a+2b+c=5
Eq3: 9a+3b+c=9

Eq2-Eq1: 3a+b=3 ...(i)
Eq3-Eq2: 5a+b=4 ...(ii)
(ii)-(i): 2a=1 → a=0.5

From (i): 3(0.5)+b=3 → 1.5+b=3 → b=1.5

From Eq1: 0.5+1.5+c=2 → 2+c=2 → c=0

So P(n) = 0.5n² + 1.5n = (n² + 3n)/2

Check:
n=1: (1+3)/2=2 ✔️
n=2: (4+6)/2=5 ✔️
n=3: (9+9)/2=9 ✔️

Given: 947 penguins

Set (n² + 3n)/2 = 947
→ n² + 3n = 1894

n² + 3n - 1894 = 0

Discriminant: 9 + 7576 = 7585

√7585 ≈ 87.08, since 87^2=7569, 88^2=7744

7585 - 7569 = 16, so not square. 87.08^2=7585? 87^2=7569, 87.1^2=7586.41, close but not integer.

87^2=7569, 7585-7569=16, so √7585 = sqrt(7569+16) not integer.

But n must be integer. Problem.

Perhaps step 3 is not 9.

Look at image: for Pattern 8, step 1: 2 penguins
Step 2: 5 penguins (arranged as 2 on bottom, 3 on top)
Step 3: should be 9? But image shows 10 or 11? Let's count.

In the third figure, it looks like a pyramid: bottom row 4, then 3, then 2, then 1? But that would be 10, but penguins are not in rows.

From the image description, it might be that each step adds a row.

Another common pattern: the number of penguins is the triangular number plus something.

Notice that 2,5,9,14,... differences 3,4,5, so P(n) = P(n-1) + (n+1)

With P(1)=2

Then P(2)=2+3=5
P(3)=5+4=9
P(4)=9+5=14
etc.

So P(n) = 2 + sum_{k=3}^{n+1} k ? Better to find closed form.

The increment is n+1 for step n? When going from n-1 to n, add n+1.

From n=1 to n=2, add 3 = 2+1
n=2 to n=3, add 4 = 3+1
so when going to step n, add n+1

So P(n) = P(1) + sum_{k=2}^n (k+1) = 2 + sum_{k=2}^n (k+1)

Sum from k=2 to n of (k+1) = sum_{j=3}^{n+1} j where j=k+1

Sum from j=3 to m of j = [m(m+1)/2] - 3, where m=n+1

Sum from 1 to m is m(m+1)/2, so from 3 to m is m(m+1)/2 - 3

So P(n) = 2 + [ (n+1)(n+2)/2 - 3 ] = (n+1)(n+2)/2 -1

Check:
n=1: (2)(3)/2 -1 = 3-1=2 ✔️
n=2: (3)(4)/2 -1 =6-1=5 ✔️
n=3: (4)(5)/2 -1=10-1=9 ✔️

So P(n) = \frac{(n+1)(n+2)}{2} - 1

Given: 947

Set \frac{(n+1)(n+2)}{2} - 1 = 947
→ \frac{(n+1)(n+2)}{2} = 948
→ (n+1)(n+2) = 1896

Let k=n+1, k(k+1)=1896

k² + k - 1896 = 0

Discriminant: 1 + 7584 = 7585

Same as before, √7585 ≈ 87.08, not integer.

87*88=7656, too big, 86*87=7482, 87*88=7656, 1896*2=3792, no.

k(k+1)=1896

Try k=43: 43*44=1892
k=44:44*45=1980

1892 < 1896 < 1980, not integer.

But 1896 - 1892 =4, so not.

Perhaps the formula is different.

Another idea: Maybe the first step is n=0.

Or perhaps in the image, step 3 has 10 penguins.

Let’s assume that for step n, it's the triangular number T(n+1) = (n+1)(n+2)/2, and for n=43, T(44)=44*45/2=990, but 947 is less.

947 +1 =948, and 948*2=1896, and 43*44=1892, 44*45=1980, no.

Perhaps it's n(n+3)/2 or other.

Let’s calculate what n gives 947 with the formula.

From P(n) = (n^2 +3n)/2 =947, n^2+3n-1894=0, discriminant 9+7576=7585, and 87^2=7569, 88^2=7744, 7585-7569=16, so not square.

But 87.08^2 = (87 + 0.08)^2 = 87^2 + 2*87*0.08 + 0.08^2 = 7569 + 13.92 + 0.0064 = 7582.9264, close to 7585, difference 2.0736, so not.

Perhaps the given number is for a different interpretation.

Let’s notice that in Pattern 1,3,6, we had n=43 for 3784,990,2838 respectively.

3784 for segments, 990 for squares, 2838 for toothpicks, all at n=43.

For Pattern 4, 173 = 4*43 +1, so n=43.

For Pattern 7, 87 = 3*29, not 43.

87 /3 =29, while others are 43.

For Pattern 2, if it were triangular, T(43)=43*44/2=946, but given 85, not match.

T(13)=91, close to 85? No.

Another thought: Perhaps for Pattern 2, the number 85 is for n=13, but 91, or maybe it's 85 for a different pattern.

Let’s list the given numbers and see if they correspond to n=43 for most.

Pattern 1: 3784 = 2*43*44 = 2*1892=3784 ✔️

Pattern 3: 990 = 44*45/2 = 990 ✔️ (for n=43 in their indexing)

Pattern 4: 173 = 4*43 +1 = 172+1=173 ✔️

Pattern 6: 2838 = 3*43*44/2 = 3*1892/2 = 3*946=2838 ✔️

Pattern 7: 87 = 3*29, not 43. 3*43=129, not 87.

87 /3 =29, so n=29 for Pattern 7.

Pattern 5: 3787 circles. If it were (n+1)^2, 61.54^2, not integer.

But 3787, and 43*88 = 3784, close to 3787.

3787 - 3784 =3, not helpful.

Perhaps for Pattern 5, the formula is different.

Let’s assume that for Pattern 5, it's also related to n=43.

What could give 3787?

Suppose C(n) = n^2 + n +1 or something.

For n=61: 61^2=3721, 3721+61=3782, +1=3783, not 3787.

n=62: 3844, too big.

3787 - 3721 =66, not related.

Another idea: Perhaps the number of circles is the same as the number of segments in Pattern 1 for n=43, which is 3784, but 3787 is given, so 3 more.

Not likely.

Perhaps it's 4n^2 +3 or something.

Let’s calculate 4*43^2 = 4*1849=7396, too big.

For n=30: 900, not.

Let’s try to see if 3787 can be expressed as a quadratic.

Suppose C(n) = an^2 + bn + c

But we need three points.

From image, assume:
n=1: 4
n=2: 9
n=3: 16 (as before)

Then C(n) = (n+1)^2, as before.

But 3787 is not square.

Unless the step is not n, but something else.

Perhaps for Pattern 5, the first figure is n=0.

Let n=0: C=4? Then n=1:9, n=2:16, so C(n) = (n+2)^2

Then for n=59: (61)^2=3721, n=60:62^2=3844, not 3787.

3787 - 3721 =66, not.

Another possibility: The shape is a hexagon or other.

I recall that in some patterns, the number of circles in a honeycomb or something.

Perhaps it's the centered hexagonal number, but those are 1,7,19,37, etc.

Not matching.

Let’s look at Pattern 8: 947 penguins.

If we assume the formula P(n) = (n+1)(n+2)/2 -1, and set to 947, then (n+1)(n+2)/2 = 948, so (n+1)(n+2) = 1896

Now, 43*44=1892, 44*45=1980, 1896 - 1892 =4, so not.

But 1896 = 43.09*44.09, not integer.

Perhaps it's n(n+1)/2 for n=43: 946, and 947 = 946 +1, so close.

For Pattern 2, if it were triangular, T(43)=946, but