Geometry Worksheet on Patterns and Reasoning | PDF | Logic ... - Free Printable

Let’s go through each pattern one by one and find the next item.

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1. 17, 23, 29, 35, 41, ...

Look at the differences:
- 23 - 17 = 6
- 29 - 23 = 6
- 35 - 29 = 6
- 41 - 35 = 6

So we’re adding 6 each time.

Next: 41 + 6 = 47

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2. 2, –4, 8, –16, 32, ...

Check the pattern:
- 2 × (–2) = –4
- –4 × (–2) = 8
- 8 × (–2) = –16
- –16 × (–2) = 32

Multiply by –2 each time.

Next: 32 × (–2) = –64

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3. 100, 81, 64, 49, ...

These are perfect squares:
- 10² = 100
- 9² = 81
- 8² = 64
- 7² = 49

Next is 6² = 36

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4. fall, winter, spring, ...

This is the order of seasons in a year:
fall → winter → spring → summer

Next: summer

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5. Spiral patterns with dots inside:

First shape: spiral with 1 dot
Second: spiral with 2 dots
Third: spiral with 3 dots

Pattern: number of dots increases by 1 each time.

Next: spiral with 4 dots

(You can draw it as a spiral with 4 small circles or dots along the curve.)

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6. Shapes: triangle (3 sides), square (4 sides), pentagon (5 sides)

Each shape has one more side than the last.

Triangle → 3 sides
Square → 4 sides
Pentagon → 5 sides
Next should be hexagon → 6 sides

Draw a regular hexagon (like a stop sign but with 6 equal sides).

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7. Dot patterns labeled 3, 6, 10...

These are triangular numbers:

- 1st: 1 dot → not shown
- 2nd: 3 dots (triangle with 2 rows: 1+2)
- 3rd: 6 dots (1+2+3)
- 4th: 10 dots (1+2+3+4)

Wait — the labels say:
- First figure: 3 dots → that’s row 2 (1+2)
- Second: 6 dots → row 3 (1+2+3)
- Third: 10 dots → row 4 (1+2+3+4)

So next is row 5: 1+2+3+4+5 = 15 dots

Arrange them in a triangle with 5 rows.

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8. Squares being divided:

First: 1 big square (no lines) → 1 part
Second: split horizontally → 2 parts
Third: split vertically too → 4 parts? Wait, let’s look again.

Actually:

- Figure 1: 1 rectangle (whole)
- Figure 2: split into 2 horizontal rectangles
- Figure 3: split into 4 smaller rectangles (2x2 grid)
- Figure 4: split into 8? No — looking closely:

Figure 1: 1 section
Figure 2: 2 sections (top/bottom)
Figure 3: 4 sections (2x2)
Figure 4: 8 sections? Actually, no — figure 4 shows 4 sections again? Wait, maybe I misread.

Looking again:

Actually, it looks like:

- Step 1: 1 box
- Step 2: divide into 2 boxes (horizontal line)
- Step 3: divide each of those into 2 → total 4 boxes (now 2x2 grid)
- Step 4: divide each of those 4 into 2? But the image shows only 4 boxes still? Hmm.

Wait — perhaps it's doubling the number of regions each time?

Step 1: 1 region
Step 2: 2 regions
Step 3: 4 regions
Step 4: 8 regions? But the fourth drawing only shows 4 regions.

Alternatively, maybe it’s dividing alternately horizontal and vertical?

But actually, looking at the images:

It seems like:

- Image 1: undivided square → 1 piece
- Image 2: divided horizontally → 2 pieces
- Image 3: divided both ways → 4 pieces
- Image 4: now each quarter is divided? But in the image, it looks like only the top-left quarter is subdivided? That would make 5 pieces? That doesn’t fit.

Wait — perhaps it’s always dividing every existing piece in half?

No — better idea: count the number of rectangles.

Image 1: 1
Image 2: 2
Image 3: 4
Image 4: 8? But visually, image 4 has 4 large rectangles, and one of them is split into 2, so total 5? That breaks the pattern.

Actually, re-examining standard problems like this — often it’s:

Each step divides all current rectangles in half.

But here’s a better way: look at how many times you’ve drawn lines.

Another approach: the number of regions doubles each time.

If step 1: 1
step 2: 2
step 3: 4
then step 4 should be 8.

But the fourth image might be misleading — perhaps it’s meant to show 8? Or maybe it’s a different pattern.

Wait — let me think differently.

In some worksheets, this pattern is:

Start with 1 square.

Then add a horizontal line → 2 rectangles.

Then add a vertical line → 4 rectangles.

Then add another horizontal line across all → 8 rectangles? But that’s not what’s drawn.

Looking at the actual image description (since I can't see it perfectly), but based on common problems:

Often, the pattern is:

- 1st: 1 region
- 2nd: 2 regions
- 3rd: 4 regions
- 4th: 8 regions

So next would be 16 regions? But that seems too big.

Wait — perhaps it’s the number of subdivisions per direction.

Alternative interpretation:

The figures show:

1. Whole square
2. Split into 2 halves (horizontally)
3. Split into 4 quarters (both directions)
4. Now each quarter is split again? But in the image, it might be showing only one quarter split — which would be inconsistent.

Actually, upon second thought — in many such worksheets, the pattern is that each new figure adds a line that cuts all existing pieces in half.

But to resolve this, let’s assume the intended pattern is doubling the number of regions each time:

1 → 2 → 4 → 8 → so next is 16? But that’s for the fifth figure.

We need the next after the fourth.

Given that:

Figure 1: 1
Figure 2: 2
Figure 3: 4
Figure 4: likely 8 (even if drawn imperfectly)

Then next should be 16? But that seems excessive.

Wait — perhaps it’s not doubling, but adding lines.

Another idea: the number of lines added.

But let’s consider a simpler pattern: the grid size.

Figure 1: 1x1
Figure 2: 2x1
Figure 3: 2x2
Figure 4: 4x2? Not matching.

I recall a similar problem where:

- Start with 1 square
- Then divide into 2
- Then divide each into 2 → 4
- Then divide each into 2 → 8

So yes, doubling.

Thus, after 4 figures, we have 8 regions, so next is 16 regions.

But drawing 16 rectangles might be messy.

Perhaps the pattern is in the number of divisions:

After n steps, number of regions = 2^(n-1)

For n=1: 2^0 = 1
n=2: 2^1 = 2
n=3: 2^2 = 4
n=4: 2^3 = 8
n=5: 2^4 = 16

So next item should be a square divided into 16 equal smaller squares (4x4 grid).

That makes sense.

So for #8, next is a 4x4 grid of squares.

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Now compiling all answers:

1. 47
2. –64
3. 36
4. summer
5. spiral with 4 dots
6. hexagon (6-sided polygon)
7. 15 dots arranged in a triangle (5 rows)
8. square divided into 16 smaller squares (4x4 grid)

Final Answer:
1. 47
2. -64
3. 36
4. summer
5. A spiral with 4 dots
6. A hexagon (6-sided shape)
7. 15 dots arranged in a triangular pattern (5 rows: 1,2,3,4,5)
8. A square divided into 16 equal smaller squares (4 by 4 grid)