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Let’s solve this Numbers Sudoku Puzzle step by step.

We have a 6x6 grid, divided into six 2x3 rectangles (each rectangle is 2 rows tall and 3 columns wide). The rules are:
- Each row must contain the numbers 1 to 6, no repeats.
- Each column must contain the numbers 1 to 6, no repeats.
- Each of the six 2x3 rectangles must also contain 1 to 6, no repeats.

Here’s the starting grid with blanks shown as underscores:

Row 1: _ 1 2 | _ 5 3
Row 2: 4 _ 5 | 6 _ 2
Row 3: _ _ 6 | _ 3 _
Row 4: 3 4 1 | 2 _ 5
Row 5: _ _ _ | 3 2 6
Row 6: 2 6 3 | 5 4 _

Let’s label the columns 1 to 6 from left to right.

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Step 1: Look at Row 1: [_, 1, 2, _, 5, 3]
Missing numbers: 4 and 6.
Check Column 1: Already has 4 (in row 2), 3 (row 4), 2 (row 6) → so can’t be 4? Wait, let’s check what’s in column 1 so far:

Column 1:
Row 1: ?
Row 2: 4
Row 3: ?
Row 4: 3
Row 5: ?
Row 6: 2
So used: 2, 3, 4 → missing: 1, 5, 6 — but row 1 already has 1 and 5, so for row 1, column 1 can only be 6? Let’s see.

Wait — better approach: look at the top-left rectangle (rows 1–2, cols 1–3):

Top-left rectangle (R1C1 to R2C3):
R1: ?, 1, 2
R2: 4, ?, 5
Numbers present: 1, 2, 4, 5 → missing: 3, 6

So the two blanks in this rectangle must be 3 and 6.

In row 1, we need to fill C1 and C4. But C4 is in the top-right rectangle.

Back to row 1: missing 4 and 6.

But in top-left rectangle, we need 3 and 6 — so one of the blanks in row 1 or row 2 in cols 1–3 must be 3 or 6.

Actually, row 1 has blanks at C1 and C4.

C1 is in top-left rect, C4 is in top-right rect.

Top-left rect needs 3 and 6. So either R1C1 or R2C2 must be 3 or 6.

But row 1 already has 1,2,5,3 → wait, row 1 has 3 at C6! So row 1 cannot have another 3. Therefore, in top-left rectangle, the 3 must go in R2C2.

Because R1C1 can’t be 3 (since row 1 already has 3 at end), so R2C2 = 3.

Then top-left rectangle missing number is 6 → must go in R1C1.

So:

R1C1 = 6
R2C2 = 3

Now update grid:

Row 1: 6 1 2 | _ 5 3
Row 2: 4 3 5 | 6 _ 2

Now row 1 missing only C4 → must be 4 (since 1,2,3,5,6 used).

So R1C4 = 4

Row 1 complete: 6 1 2 | 4 5 3

Row 2: missing C5 → numbers used: 4,3,5,6,2 → missing 1 → so R2C5 = 1

Row 2 complete: 4 3 5 | 6 1 2

Now top-right rectangle (R1–2, C4–6):
R1: 4,5,3
R2: 6,1,2 → all good, has 1-6.

Now move to middle-left rectangle (R3–4, C1–3)

Current:

Row 3: _ _ 6
Row 4: 3 4 1

So cells: R3C1, R3C2, R3C3=6; R4C1=3, R4C2=4, R4C3=1

Used in this rectangle: 6,3,4,1 → missing: 2,5

So R3C1 and R3C2 must be 2 and 5.

Look at column 1:

Col 1 so far:
R1:6, R2:4, R3:?, R4:3, R5:?, R6:2 → used: 6,4,3,2 → missing: 1,5

But row 3 needs 2 or 5 for C1 — but 2 is already in col 1 (R6), so R3C1 cannot be 2 → must be 5.

Then R3C2 = 2

So:

Row 3: 5 2 6 | _ 3 _
Row 4: 3 4 1 | 2 _ 5

Now row 3 missing C4 and C6.

Numbers used in row 3: 5,2,6,3 → missing: 1,4

Check column 4:

Col 4:
R1:4, R2:6, R3:?, R4:2, R5:3, R6:5 → used: 4,6,2,3,5 → missing: 1

So R3C4 must be 1

Then R3C6 = 4 (only one left for row 3)

Row 3 complete: 5 2 6 | 1 3 4

Now row 4: 3 4 1 | 2 _ 5 → missing C5

Used: 3,4,1,2,5 → missing 6 → R4C5 = 6

Row 4 complete: 3 4 1 | 2 6 5

Middle-right rectangle (R3–4, C4–6):
R3:1,3,4
R4:2,6,5 → has 1-6, good.

Now bottom-left rectangle (R5–6, C1–3)

Row 5: _ _ _
Row 6: 2 6 3

So cells: R5C1, R5C2, R5C3; R6C1=2, R6C2=6, R6C3=3

Used: 2,6,3 → missing: 1,4,5

Row 5: all three blank.

Also, row 5 will have C4=3, C5=2, C6=6 → so row 5: [?,?,?, 3,2,6]

So numbers used in row 5: 3,2,6 → missing: 1,4,5 → matches.

Now look at columns.

Column 1:
R1:6, R2:4, R3:5, R4:3, R5:?, R6:2 → used: 6,4,5,3,2 → missing: 1 → so R5C1 = 1

Column 2:
R1:1, R2:3, R3:2, R4:4, R5:?, R6:6 → used: 1,3,2,4,6 → missing: 5 → R5C2 = 5

Then R5C3 must be 4 (only one left for row 5)

Row 5: 1 5 4 | 3 2 6

Bottom-left rectangle: R5:1,5,4; R6:2,6,3 → has 1-6, good.

Now last cell: R6C6

Row 6: 2 6 3 | 5 4 _ → used: 2,6,3,5,4 → missing: 1 → R6C6 = 1

Check column 6:

Col 6:
R1:3, R2:2, R3:4, R4:5, R5:6, R6:1 → 1-6, good.

Final grid:

Row 1: 6 1 2 | 4 5 3
Row 2: 4 3 5 | 6 1 2
Row 3: 5 2 6 | 1 3 4
Row 4: 3 4 1 | 2 6 5
Row 5: 1 5 4 | 3 2 6
Row 6: 2 6 3 | 5 4 1

Double-check each row, column, and rectangle — all have 1-6 with no repeats.

Final Answer:
The completed puzzle is:

6 1 2 4 5 3
4 3 5 6 1 2
5 2 6 1 3 4
3 4 1 2 6 5
1 5 4 3 2 6
2 6 3 5 4 1