Identify number patterns and fill in missing numbers in this printable math worksheet.

Let's solve each of these "Two-Rule Pattern" problems step by step. The key is to identify two alternating rules (usually one for odd positions and one for even positions, or a repeating pattern of operations) that generate the sequence.

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Problem 1: 5, 3, 9, 2, 13, __, __, __, __

- Position: 1: 5, 2: 3, 3: 9, 4: 2, 5: 13
- Let's separate odd and even positions:
- Odd positions (1st, 3rd, 5th): 5, 9, 13 → increases by 4 each time.
- So next odd position (7th): 13 + 4 = 17
- Then 9th: 17 + 4 = 21
- Even positions (2nd, 4th): 3, 2 → decreases by 1.
- Next even (6th): 2 - 1 = 1
- Then 8th: 1 - 1 = 0
- So the sequence becomes:
- 5, 3, 9, 2, 13, 1, 17, 0, 21

✔ Answer: 1, 17, 0, 21

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Problem 2: 24, 23, 25, 26, 26, __, __, __, __

- Sequence: 24, 23, 25, 26, 26
- Let’s look at differences:
- 24 → 23: -1
- 23 → 25: +2
- 25 → 26: +1
- 26 → 26: 0
- Not obvious. Try separating into two interleaved sequences:
- Odd positions: 24, 25, 26 → increasing by 1
- So next odd (7th): 27, then 9th: 28
- Even positions: 23, 26, ? → 23 to 26 = +3
- So next even (6th): 26 + 3 = 29, then 8th: 32?
- But wait: only 23, 26 so far — maybe not yet clear.
- Wait — let's list positions:

| Pos | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
|-----|---|---|---|---|---|---|---|---|---|
| Val |24|23|25|26|26|? |? |? |? |

Odd positions: 1:24, 3:25, 5:26 → increasing by 1 → so 7:27, 9:28
Even positions: 2:23, 4:26, 6:?, 8:?, 10:?

From 23 to 26: +3 → maybe constant +3? So:
- 6th: 26 + 3 = 29
- 8th: 29 + 3 = 32

So sequence: 24, 23, 25, 26, 26, 29, 27, 32, 28

Wait — but 7th is odd → should be 27, 8th even → 32, 9th odd → 28

But check: 5th is 26, 7th should be 27, 9th 28 — yes.

So missing numbers: 6th: 29, 7th: 27, 8th: 32, 9th: 28

✔ Answer: 29, 27, 32, 28

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Problem 3: 14, 17, 13, 15, 12, __, __, __, __

- Positions:
- 1:14, 2:17, 3:13, 4:15, 5:12
- Odd positions: 1:14, 3:13, 5:12 → decreasing by 1 → next odd (7th): 11, then 9th:10
- Even positions: 2:17, 4:15 → decreasing by 2 → next even (6th): 13, then 8th:11
- So:
- 6th (even): 13
- 7th (odd): 11
- 8th (even): 11
- 9th (odd): 10

Sequence: 14, 17, 13, 15, 12, 13, 11, 11, 10

✔ Answer: 13, 11, 11, 10

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Problem 4: 74, 70, 77, 75, 80, __, __, __, __

- Positions:
- 1:74, 2:70, 3:77, 4:75, 5:80
- Odd positions: 74, 77, 80 → increasing by 3 → next: 83, 86
- Even positions: 70, 75 → increasing by 5 → next: 80, 85
- So:
- 6th (even): 80
- 7th (odd): 83
- 8th (even): 85
- 9th (odd): 86

Sequence: 74, 70, 77, 75, 80, 80, 83, 85, 86

✔ Answer: 80, 83, 85, 86

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Problem 5: 32, 36, 31, 38, 30, __, __, __, __

- Positions:
- 1:32, 2:36, 3:31, 4:38, 5:30
- Odd positions: 32, 31, 30 → decreasing by 1 → next: 29, 28
- Even positions: 36, 38 → increasing by 2 → next: 40, 42
- So:
- 6th (even): 40
- 7th (odd): 29
- 8th (even): 42
- 9th (odd): 28

Sequence: 32, 36, 31, 38, 30, 40, 29, 42, 28

✔ Answer: 40, 29, 42, 28

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Problem 6: 1, 10, 3, 12, 5, __, __, __, __

- Positions:
- 1:1, 2:10, 3:3, 4:12, 5:5
- Odd positions: 1, 3, 5 → increasing by 2 → next: 7, 9
- Even positions: 10, 12 → increasing by 2 → next: 14, 16
- So:
- 6th (even): 14
- 7th (odd): 7
- 8th (even): 16
- 9th (odd): 9

Sequence: 1, 10, 3, 12, 5, 14, 7, 16, 9

✔ Answer: 14, 7, 16, 9

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Problem 7: 56, 52, 54, 55, 52, __, __, __, __

- Positions:
- 1:56, 2:52, 3:54, 4:55, 5:52
- Odd positions: 56, 54, 52 → decreasing by 2 → next: 50, 48
- Even positions: 52, 55 → increasing by 3 → next: 58, 61
- So:
- 6th (even): 58
- 7th (odd): 50
- 8th (even): 61
- 9th (odd): 48

Sequence: 56, 52, 54, 55, 52, 58, 50, 61, 48

✔ Answer: 58, 50, 61, 48

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Problem 8: 17, 21, 13, 23, 9, __, __, __, __

- Positions:
- 1:17, 2:21, 3:13, 4:23, 5:9
- Odd positions: 17, 13, 9 → decreasing by 4 → next: 5, 1
- Even positions: 21, 23 → increasing by 2 → next: 25, 27
- So:
- 6th (even): 25
- 7th (odd): 5
- 8th (even): 27
- 9th (odd): 1

Sequence: 17, 21, 13, 23, 9, 25, 5, 27, 1

✔ Answer: 25, 5, 27, 1

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Problem 9: 90, 10, 85, 15, 80, __, __, __, __

- Positions:
- 1:90, 2:10, 3:85, 4:15, 5:80
- Odd positions: 90, 85, 80 → decreasing by 5 → next: 75, 70
- Even positions: 10, 15 → increasing by 5 → next: 20, 25
- So:
- 6th (even): 20
- 7th (odd): 75
- 8th (even): 25
- 9th (odd): 70

Sequence: 90, 10, 85, 15, 80, 20, 75, 25, 70

✔ Answer: 20, 75, 25, 70

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Problem 10: 67, 70, 64, 68, 61, __, __, __, __

- Positions:
- 1:67, 2:70, 3:64, 4:68, 5:61
- Odd positions: 67, 64, 61 → decreasing by 3 → next: 58, 55
- Even positions: 70, 68 → decreasing by 2 → next: 66, 64
- So:
- 6th (even): 66
- 7th (odd): 58
- 8th (even): 64
- 9th (odd): 55

Sequence: 67, 70, 64, 68, 61, 66, 58, 64, 55

✔ Answer: 66, 58, 64, 55

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✔ Final Answers Summary:

| Problem | Missing Numbers |
|--------|------------------|
| 1 | 1, 17, 0, 21 |
| 2 | 29, 27, 32, 28 |
| 3 | 13, 11, 11, 10 |
| 4 | 80, 83, 85, 86 |
| 5 | 40, 29, 42, 28 |
| 6 | 14, 7, 16, 9 |
| 7 | 58, 50, 61, 48 |
| 8 | 25, 5, 27, 1 |
| 9 | 20, 75, 25, 70 |
| 10 | 66, 58, 64, 55 |

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💡 Pattern Strategy Recap:
- Most patterns alternate between odd and even positions.
- Look for two separate arithmetic sequences: one for odd-indexed terms, one for even-indexed terms.
- Identify the rule (addition/subtraction) in each subsequence.

Let me know if you'd like this as a printable answer sheet!