purlJam

<planning>
1. **Design Strategy**:
   - User wants a basic sweater with V-neck, bottom hem (ribbing), and cuffs (ribbing)
   - Beginner difficulty level — keep instructions clear, avoid complex techniques
   - Using the Fitted Set-In Pullover template as the base
   - V-neck modification: Front neck shaping begins earlier than crew neck (approximately 5-6" before shoulders), with gradual decreases on each side creating the V shape
   - Yarn weight: Worsted
   - Standard worsted gauge: 18 sts × 24 rows = 4" (10 cm) in stockinette on US 8 / 5 mm needles
   - Ribbing needle: US 7 / 4.5 mm (one size down for tighter ribbing)
   - Bottom ribbing and cuffs: k2p2 rib, 2" for cuffs, 2.5" for bottom hem — simple and beginner-friendly
   - Sizes: S(34-36"), M(38-40"), L(42-44"), XL(46-48"), 2XL(50-52")
   - Positive ease: ~2" for a comfortable, slightly fitted fit appropriate for a beginner sweater
   - Finished chest: S=36", M=40", L=44", XL=48", 2XL=52"

2. **Construction Plan**:
   - Work flat in pieces: Back, Front, two Sleeves
   - Bottom up construction
   - Back piece: Full width, set-in armhole shaping, shoulder shaping
   - Front piece: Same as back until armhole, then V-neck shaping begins simultaneously or shortly after armhole shaping
   - Sleeves: Worked from cuff up, tapered increases, then set-in sleeve cap shaping
   - Assembly: Seam shoulders, set in sleeves, seam sides and sleeve underarms
   - V-neck finishing: Pick up stitches around V-neck opening and work k1p1 or k2p2 rib in the round (or flat with seam at center back) — for beginners, I'll use k1p1 rib on smaller needles, worked flat and seamed, or picked up in round. Actually, for a V-neck, picking up in the round with a center front decrease is the standard clean method. I'll do picked-up neckband worked in the round with a center front decrease at the V point.

3. **Quality Assurance**:
   - V-neck shaping: The V-neck opens from a center stitch split. For set-in construction, the V typically starts about 7-8" below the shoulder (deeper than crew neck). For worsted gauge at 24 rows/4", that's approximately 42-48 rows before bind-off.
   - Need to split front into two halves at center and work each side separately
   - Neck decreases: Every RS row (every other row) for steeper V, or every 4 rows for more gradual V — for a classic V-neck depth of ~7-8" below shoulder, working decreases every other row is appropriate
   - Ribbing divisibility: Cast-on must be divisible by 4 for k2p2 rib
   - Sleeve cap shaping: Standard set-in cap — bind off same as armhole at start, then decrease every RS row, then bind off remaining stitches
   - Center front V-neck: Work a centered double decrease (sl2-k1-p2sso) at the V point of the neckband, or work one decrease each side of center stitch
   - For beginner level, I'll keep the neckband instructions very clear

4. **Final Calculations**:

**Gauge**: 18 sts × 24 rows = 4" in stockinette on US 8 / 5 mm needles
**Sts per inch**: 18/4 = 4.5 sts per inch
**Rows per inch**: 24/4 = 6 rows per inch

**Finished Chest Measurements** (with ~2" ease):
- S: 36" → half width = 18"
- M: 40" → half width = 20"
- L: 44" → half width = 22"
- XL: 48" → half width = 24"
- 2XL: 52" → half width = 26"

**Back/Front stitch counts** (full width, cast on for BACK = half chest × 2 = full chest width... wait):
Actually for a pullover worked flat in pieces, the BACK piece = half the total chest circumference.
- S: 18" × 4.5 sts/in = 81 sts → round to 80 (divisible by 4 for k2p2 rib)
- M: 20" × 4.5 = 90 sts → 92 (divisible by 4) — actually let me check: 90/4 = 22.5, so round to 92
- L: 22" × 4.5 = 99 sts → 100 (divisible by 4)
- XL: 24" × 4.5 = 108 sts → 108 (divisible by 4 ✓)
- 2XL: 26" × 4.5 = 117 sts → 116 (divisible by 4) or 120

Let me reconsider. Let me be more precise:
- S: 18" × 4.5 = 81 → 80 sts (divisible by 4 ✓)
- M: 20" × 4.5 = 90 → 92 sts (divisible by 4 ✓)  — wait, 90 IS divisible by... 90/4 = 22.5, no. So 88 or 92. 92 is closer to 90. Let me use 92.
- L: 22" × 4.5 = 99 → 100 sts (divisible by 4 ✓)
- XL: 24" × 4.5 = 108 → 108 sts (divisible by 4 ✓)
- 2XL: 26" × 4.5 = 117 → 116 sts (divisible by 4 ✓) — 116/4 = 29 ✓

Actual finished widths:
- S: 80/4.5 = 17.8" → ×2 = 35.6" ≈ 36" ✓
- M: 92/4.5 = 20.4" → ×2 = 40.9" ≈ 41" ✓ (close enough)
- L: 100/4.5 = 22.2" → ×2 = 44.4" ≈ 44" ✓
- XL: 108/4.5 = 24" → ×2 = 48" ✓
- 2XL: 116/4.5 = 25.8" → ×2 = 51.6" ≈ 52" ✓

Good. Let me reconsider M — 92 gives 40.9". The target is 40" finished. Let me use 88 instead:
- M: 88/4.5 = 19.6" → ×2 = 39.1" ≈ 39". Hmm, that's a bit small. 

Actually, for a worsted weight sweater with 4.5 sts/inch, having the stitch count be exact to 4 is fine because gauge can vary slightly. Let me keep 92 for M since it gives a better fit for M (38-40" body + 2" ease = 40-42" finished).

Wait, let me reconsider ease more carefully:
- S: body 34-36", finished chest 38" (2-4" ease) → 38" / 2 = 19" per piece → 19 × 4.5 = 85.5 → 84 sts (div by 4)
Hmm, this is getting complicated. Let me simplify by using standard sweater sizing with standard ease:

Standard worsted sweater, ~2" positive ease:
- S: body 34-36", finished 38" → per piece 19" → 86 sts... not div by 4... → 88 sts = 22", finished 44". Too much.

Let me try a different approach. I'll define finished chest, compute sts, and round to nearest multiple of 4:

Finished bust/chest:
- S: 38"  → per piece 19" → 19 × 4.5 = 85.5 → 84 sts → actual 84/4.5 = 18.67" × 2 = 37.3" (rounds to ~37")
- M: 42" → per piece 21" → 21 × 4.5 = 94.5 → 96 sts → 96/4.5 = 21.3" × 2 = 42.7" ✓
- L: 46" → per piece 23" → 23 × 4.5 = 103.5 → 104 sts → 104/4.5 = 23.1" × 2 = 46.2" ✓
- XL: 50" → per piece 25" → 25 × 4.5 = 112.5 → 112 sts → 112/4.5 = 24.9" × 2 = 49.8" ✓
- 2XL: 54" → per piece 27" → 27 × 4.5 = 121.5 → 120 sts → 120/4.5 = 26.7" × 2 = 53.3" ✓

Hmm, S is off. Let me try 88 for S: 88/4.5 = 19.6" × 2 = 39.1" ≈ 39". With body size 34-36", 39" finished gives ~3-5" ease. Still okay for a beginner/casual sweater.

Actually, let me just go with round numbers for ease of explanation:

Let me restart with clear targets:

**Target finished chest circumference** (body measurement + ~2-4" ease):
- S (34-36" body): 38" finished
- M (38-40" body): 42" finished  
- L (42-44" body): 46" finished
- XL (46-48" body): 50" finished
- 2XL (50-52" body): 54" finished

**Stitches per piece** (half circumference):
- S: 19" × 4.5 = 85.5 → **88 sts** (actual: 88/4.5 = 19.6" × 2 = 39.1" ≈ 39")
- M: 21" × 4.5 = 94.5 → **96 sts** (actual: 96/4.5 = 21.3" × 2 = 42.7" ≈ 43")
- L: 23" × 4.5 = 103.5 → **104 sts** (actual: 104/4.5 = 23.1" × 2 = 46.2" ≈ 46")
- XL: 25" × 4.5 = 112.5 → **112 sts** (actual: 112/4.5 = 24.9" × 2 = 49.8" ≈ 50")
- 2XL: 27" × 4.5 = 121.5 → **120 sts** (actual: 120/4.5 = 26.7" × 2 = 53.3" ≈ 53")

All divisible by 4? 
- 88/4 = 22 ✓
- 96/4 = 24 ✓
- 104/4 = 26 ✓
- 112/4 = 28 ✓
- 120/4 = 30 ✓

**Body length** (armhole to bottom):
Standard lengths for worsted weight sweater:
- S/M: 14" to underarm
- L/XL/2XL: 15" to underarm
Let me standardize: 14.5" for all sizes, or use variable:
- S: 14"
- M: 14"  
- L: 15"
- XL: 15"
- 2XL: 15"

Actually let me use: S/M = 14", L/XL/2XL = 15"

**Armhole shaping** (bind offs):
For set-in sleeves, bind off at underarm:
- S: bind off 4 sts each side (2× BO)
- M: bind off 4 sts each side
- L: bind off 5 sts each side
- XL: bind off 5 sts each side
- 2XL: bind off 6 sts each side

Then additional decreases to shape armhole:
Typically: dec 1 st each side every RS row 3(4,4,5,5) times

Stitches after armhole shaping:
- S: 88 - 8 - 6 = 74 sts
- M: 96 - 8 - 8 = 80 sts
- L: 104 - 10 - 8 = 86 sts
- XL: 112 - 10 - 10 = 92 sts
- 2XL: 120 - 12 - 10 = 98 sts

Let me recalculate:
- S: 88 - (4×2) - (3×2) = 88 - 8 - 6 = 74 sts ✓
- M: 96 - (4×2) - (4×2) = 96 - 8 - 8 = 80 sts ✓
- L: 104 - (5×2) - (4×2) = 104 - 10 - 8 = 86 sts ✓
- XL: 112 - (5×2) - (5×2) = 112 - 10 - 10 = 92 sts ✓
- 2XL: 120 - (6×2) - (5×2) = 120 - 12 - 10 = 98 sts ✓

**Armhole depth**: 
- S/M: 7.5"
- L/XL: 8"
- 2XL: 8.5"

**Shoulder width** and **back neck width**:
Standard back neck width: ~6.5-7" for all sizes
Back neck sts: 6.5" × 4.5 = 29.25 → 30 sts (for M, adjust proportionally)

Actually, let me use a consistent back neck width:
- S: 28 sts (6.2")
- M: 30 sts (6.7")
- L: 32 sts (7.1")
- XL: 34 sts (7.6")
- 2XL: 36 sts (8.0")

Shoulder sts per side (after armhole shaping):
- S: (74 - 28) / 2 = 23 sts per shoulder
- M: (80 - 30) / 2 = 25 sts per shoulder
- L: (86 - 32) / 2 = 27 sts per shoulder
- XL: (92 - 34) / 2 = 29 sts per shoulder
- 2XL: (98 - 36) / 2 = 31 sts per shoulder

**Shoulder bind-off** (3-step for gradual shaping):
Divide each shoulder into 3 roughly equal parts:
- S: 23 sts → 8, 8, 7 (or 8, 7, 8)
- M: 25 sts → 9, 8, 8
- L: 27 sts → 9, 9, 9
- XL: 29 sts → 10, 10, 9
- 2XL: 31 sts → 11, 10, 10

**Total body length** = bottom ribbing + stockinette body + armhole depth:
Ribbing = 2.5"
Body to underarm (stockinette) = S/M: 14", L/XL/2XL: 15"
Armhole depth = S/M: 7.5", L: 8", XL/2XL: 8.5"
Total length from CO to shoulder:
- S: 2.5 + 14 + 7.5 = 24"
- M: 2.5 + 14 + 7.5 = 24"
- L: 2.5 + 15 + 8 = 25.5"
- XL: 2.5 + 15 + 8.5 = 26"
- 2XL: 2.5 + 15 + 8.5 = 26"

**V-NECK SHAPING**:
V-neck depth: typically 7-8" below shoulder seam. For a set-in pullover, the V-neck shaping on the front begins about 7-8" before the shoulder bind-off.

For S/M with 7.5" armhole: The V-neck would ideally start at or just before the armhole (7-8" before shoulder). Let's say V-neck starts at same row as armhole shaping begins for smaller sizes, or 1" before for larger sizes. This gives a nice deep V.

Actually, for a clean beginner pattern, let me have the V-neck shaping begin AT the underarm — this simplifies the instructions because both pieces of shaping start on the same row.

V-neck depth = armhole depth (since both start at same row):
- S/M: ~7.5" deep V
- L: ~8" deep V
- XL/2XL: ~8.5" deep V

This is a good, classic V-neck depth.

**V-neck shaping**:
At the underarm row, split front stitches at center:
- S: 74 sts after armhole BO on first row — wait, I need to think about this more carefully.

The front piece has the same cast-on as the back. At the underarm, we bind off the same underarm stitches. The remaining stitches on the front = same as back at that point.

After initial armhole BO row (both sides), front sts remaining:
- S: 88 - 8 = 80 sts (before dec rows) — actually on the bind-off row we bind off 4 sts at each side, so:
  Row 1 (RS): BO 4, knit to end = 84 sts
  Row 2 (WS): BO 4, purl to end = 80 sts

Hmm, but for the V-neck split, I want to divide at center. After the armhole BOs on both sides, I have:
- S: 80 sts → split into 40 + 40 for V-neck

For the V-neck, I'll work each side of the neck separately. On the same RS row as the first armhole dec row (after the bind-off rows), I'll split at center and begin neck decreases.

Let me reconsider the sequencing for the front:

**FRONT** (setting up V-neck AND armhole simultaneously):

Setup: At the start of armhole shaping (same point as on Back), we split the front for the V-neck.

For S (88 sts):
- Row 1 (RS): BO 4 sts for armhole, k to center, place marker or note center, join second ball of yarn and k to end, BO 4 sts... 

Actually this gets complicated for beginners. Better approach: BO for armhole on both sides first (2 rows), THEN split at center for V-neck, working each side separately.

Alternative (cleaner for beginners): 
Row 1 (RS): BO 4 sts, k to 2 sts before center, k2tog, pm, ssk, k to end — no, that's too complex.

Simplest approach for beginners:
After binding off for armholes (same as back, 2 rows), then on next RS row, split at center:
Right Front: Work to center, leave sts for left on holder
Work right front with neck dec (ssk at neck edge on RS rows) and armhole dec simultaneously for a few rows, then just neck decs

Let me work through S size in detail:
CO: 88 sts
After ribbing and body, at underarm (80 sts... wait no, 88 sts still at underarm):

**Back underarm**: 88 sts → BO 4 at beg of next 2 rows = 80 sts → then dec 1 each side every RS row × 3 = 74 sts

**Front underarm**: Same bind-offs as back:
Row 1 (RS): BO 4, k to end = 84 sts
Row 2 (WS): BO 4, p to end = 80 sts

Now split for V-neck on next RS row:
80 sts ÷ 2 = 40 sts per side

RIGHT FRONT (neck edge at left when working RS):
Row 3 (RS-dec): k2tog (armhole dec), k to last 2 sts, ssk (neck dec) — wait, which edge is neck?

Let me think about orientation:
- Right front (as worn) = right side of your body
- When working right front on RS, you're working from right to left on the page, but knitting left to right
- For right front: armhole is at the right edge (start of RS row), neck is at left edge (end of RS row)
- Armhole dec (RS): k2, ssk (decrease at armhole edge) or: k1, ssk
- Neck dec (RS): k to last 3 sts, k2tog, k1

Actually for cleaner edges:
- Armhole edge dec (RS): k1, ssk, work to end
- Neck edge dec (RS): work to last 3 sts, k2tog, k1

For left front (mirror):
- Armhole edge dec (RS, at end of RS row): work to last 3 sts, k2tog, k1
- Neck edge dec (RS, at start): k1, ssk, work to end

Let me just calculate how many neck decreases are needed:

For S: 
- After split: 40 sts per side
- Armhole decs: 3 times (same as back) → removes 3 sts per side from armhole edge
- Remaining after armhole decs: 40 - 3 = 37 sts per side (armhole) 

Wait, I need to reconsider. The 40 sts per side already includes 4 sts removed from the initial BO. Let me recount:

88 sts cast on for front.
After armhole BOs (4+4 = 8 sts removed): 80 sts remain
Split at center: 40 sts per side

Each 40-st half includes:
- The armhole edge on one side
- The neck edge (center) on the other side

We need the shoulder to end up with 23 sts (same as back).
After armhole shaping (3 decs on armhole side): 40 - 3 = 37 sts per side
Neck decreases needed: 37 - 23 = 14 decs per side

V-neck depth = 7.5" armhole depth = 45 rows
We need 14 neck decreases over 45 rows.
If we dec every other RS row (every 4 rows): 14 × 4 = 56 rows — too many
If we dec every RS row (every 2 rows): 14 × 2 = 28 rows — only 28 of 45 rows used, leaves 17 rows straight

Option: Dec every RS row for 14 rows (28 rows total), then work even to shoulder.
OR: Dec every RS row × 8, then every other RS row × 6 = 8 + 6 = 14 decs, using 16 + 24 = 40 rows + 5 rows even = 45 rows. 

For a beginner pattern, simplest: dec every RS row (every 2 rows) × 14 = uses 28 rows, then work even for remaining rows. Let's check: 28 rows / 6 rows per inch = 4.7". Then 45 - 28 = 17 rows even = 17/6 = 2.8". Total = 7.5" ✓

Actually I realize the armhole decs also happen in the first 6 rows (3 RS rows), so they overlap with neck decs. Let me clarify that the neck dec starts AFTER we split (from the split row), and armhole decs happen at armhole edge simultaneously.

For beginner simplicity, I'll state: after splitting, dec 1 st at NECK edge on every RS row × 14 times (for S), simultaneously dec at armhole edge on first 3 RS rows after split. After 14 neck decs, work even to shoulder. 

Hmm, this is getting complex. Let me simplify by having the armhole shaping identical on front and back (so knitters work it the same way), and just add neck decreases on the front.

Let me redo this more carefully for all sizes:

**Stitches per half of front after initial armhole BOs**:
- S: 80/2 = 40 sts each side
- M: (96-8)/2 = 88/2 = 44 sts each side
- L: (104-10)/2 = 94/2 = 47 sts each side
- XL: (112-10)/2 = 102/2 = 51 sts each side
- 2XL: (120-12)/2 = 108/2 = 54 sts each side

**Remaining armhole decs (same as back)**:
- S: 3 per side
- M: 4 per side
- L: 4 per side
- XL: 5 per side
- 2XL: 5 per side

**Sts after all armhole shaping** (= should match back):
- S: 40-3=37 per side
- M: 44-4=40 per side
- L: 47-4=43 per side
- XL: 51-5=46 per side
- 2XL: 54-5=49 per side

**Shoulder sts needed** (same as back):
- S: 23
- M: 25
- L: 27
- XL: 29
- 2XL: 31

**Neck decreases needed per side** (= sts after armhole - shoulder sts):
- S: 37-23=14 neck decs
- M: 40-25=15 neck decs
- L: 43-27=16 neck decs
- XL: 46-29=17 neck decs
- 2XL: 49-31=18 neck decs

**Armhole depth in rows** (rows available for neck shaping):
- S/M: 7.5" × 6 = 45 rows
- L: 8" × 6 = 48 rows
- XL/2XL: 8.5" × 6 = 51 rows

**Can we do dec every RS row (every 2 rows)?**
- S: 14 decs × 2 rows = 28 rows < 45 rows ✓ (then 17 rows even)
- M: 15 × 2 = 30 < 45 ✓ (then 15 rows even)
- L: 16 × 2 = 32 < 48 ✓ (then 16 rows even)
- XL: 17 × 2 = 34 < 51 ✓ (then 17 rows even)
- 2XL: 18 × 2 = 36 < 51 ✓ (then 15 rows even)

Great! All sizes work with dec every RS row. For a beginner pattern, this is clear and simple.

**Check: sts remaining on front after neck decs (at shoulder)**:
After all armhole decs AND all neck decs:
- S: 40 - 3 - 14 = 23 ✓ (matches back shoulder)
- M: 44 - 4 - 15 = 25 ✓
- L: 47 - 4 - 16 = 27 ✓
- XL: 51 - 5 - 17 = 29 ✓
- 2XL: 54 - 5 - 18 = 31 ✓

**SLEEVE CALCULATIONS**:

Cuff circumference (tight, for fitted sleeve):
- S: 9" → 9 × 4.5 = 40.5 → **40 sts** (div by 4 ✓)
- M: 10" → 10 × 4.5 = 45 → **44 sts** (div by 4 ✓)
- L: 11" → 11 × 4.5 = 49.5 → **48 sts** (div by 4... 48/4=12 ✓)
- XL: 11.5" → 11.5 × 4.5 = 51.75 → **52 sts** (div by 4 ✓)
- 2XL: 12" → 12 × 4.5 = 54 → **52 sts** (div by 4 ✓) — or 56?

Let me reconsider: I want a clear graduated cuff. Let me use:
- S: 40 sts (actual: 40/4.5 = 8.9" ≈ 9")
- M: 44 sts (actual: 44/4.5 = 9.8" ≈ 10")
- L: 48 sts (actual: 48/4.5 = 10.7" ≈ 10.75")
- XL: 52 sts (actual: 52/4.5 = 11.6" ≈ 11.5")
- 2XL: 56 sts (actual: 56/4.5 = 12.4" ≈ 12.5")

All divisible by 4 ✓

**Sleeve width at underarm** (should match armhole width on body):
The sleeve cap needs to fit the armhole opening.
Armhole opening = underarm BO + dec rows
Let me calculate armhole opening circumference:
Armhole opening depth ≈ each side's total BO + decs:
- S: 4 + 3 = 7 sts per side × 2 = 14 sts removed from body width
- Actually the "armhole opening" for sleeve fitting is not directly derived this way.

For a set-in sleeve, the sleeve width at underarm should equal approximately the chest width minus the armhole shaping:
Standard sleeve cap height ≈ armhole depth - 0.5-1" = 6.5-7" for S/M, 7-7.5" for L-2XL

Sleeve underarm width (number of sts at widest before cap):
Standard measurement: about 12-14" for S, 14-16" for M, 15-17" for L, etc.
Let me use:
- S: 13" → 13 × 4.5 = 58.5 → **58 sts** — hmm, want even number for symmetry, 58 is even ✓
- M: 14.5" → 14.5 × 4.5 = 65.25 → **64 sts** (even ✓)  
- L: 16" → 16 × 4.5 = 72 → **72 sts** (even ✓)
- XL: 17" → 17 × 4.5 = 76.5 → **76 sts** (even ✓)
- 2XL: 18" → 18 × 4.5 = 81 → **80 sts** (even ✓)

**Sleeve increases**:
From cuff sts to underarm sts, increasing 1 st each side (k/f/b or m1 at each end):
- S: 58-40 = 18 sts to add → 9 increase rows (1 st each end)
- M: 64-44 = 20 sts → 10 increase rows
- L: 72-48 = 24 sts → 12 increase rows
- XL: 76-52 = 24 sts → 12 increase rows
- 2XL: 80-56 = 24 sts → 12 increase rows

**Sleeve length** (from cuff to underarm, after 2" ribbing):
Standard sleeve lengths for set-in:
- S: 17" total (including 2" cuff ribbing) → 15" to work after ribbing
- M: 17.5" total → 15.5"
- L: 18" total → 16"
- XL: 18.5" total → 16.5"
- 2XL: 19" total → 17"

Rows from after ribbing to underarm (using 6 rows/inch):
- S: 15" × 6 = 90 rows
- M: 15.5" × 6 = 93 rows
- L: 16" × 6 = 96 rows
- XL: 16.5" × 6 = 99 rows
- 2XL: 17" × 6 = 102 rows

**Increase frequency**:
Total increases needed / total rows:
- S: 9 increases over 90 rows → every 10 rows (every 10th row)
- M: 10 increases over 93 rows → every ~9 rows → let's say every 9th row... but I want RS rows ideally: every 10 rows is every 5th RS row approximately... 

Actually, sleeve increases are typically done every 6th, 8th, or 10th row (always on RS). Let me calculate:

- S: 9 incs, 90 rows → every 10 rows ✓ (9 × 10 = 90) ✓
- M: 10 incs, 93 rows → every 9 rows? (10 × 9 = 90, leaving 3 rows) → say every 9 rows, 10 times = 90 rows, then 3 rows even. Fine.
- Actually let me round: every 10 rows for M too: 10 × 10 = 100 > 93. Too many.
  Every 8 rows: 10 × 8 = 80 rows + 13 rows even = 93. OK, but increases end 13 rows before underarm.
  Every 9 rows: 10 × 9 = 90 + 3 rows even = 93. ✓

- L: 12 incs, 96 rows → every 8 rows: 12 × 8 = 96 ✓ exactly
- XL: 12 incs, 99 rows → every 8 rows: 12 × 8 = 96, + 3 rows even. Or every 8 rows × 12 = 96 rows, 3 rows even.
- 2XL: 12 incs, 102 rows → every 8 rows: 12 × 8 = 96, + 6 rows even. Fine.

Summary of increase frequency:
- S: every 10 rows × 9 times
- M: every 9 rows × 10 times (+ 3 rows even at end)
- L: every 8 rows × 12 times
- XL: every 8 rows × 12 times (+ 3 rows even)
- 2XL: every 8 rows × 12 times (+ 6 rows even)

Hmm, the "3 rows even" is a bit awkward. Let me adjust the sleeve lengths or increase frequencies slightly:

For M: 10 incs, let me try every 10 rows: 10 × 10 = 100 rows. That means the sleeve after ribbing is 100 rows = 100/6 = 16.7" ≈ 16.75". Total with ribbing = 18.75". A bit long. OK for taller people.
OR: Do first few increases every 8 rows and later every 10: mixed. Too complex for beginners.
BETTER: Just use "every 10 rows" and note that sleeve can be adjusted.

Actually let me just use every 10 rows for S/M and every 8 rows for L/XL/2XL, and adjust total sleeve length to match:

- S: every 10 rows × 9 = 90 rows = 15" to underarm (+2" cuff = 17" total) ✓
- M: every 10 rows × 10 = 100 rows = 16.7" to underarm (+2" = 18.7"). Hmm, long. Let me try every 8 rows × 10 = 80 rows = 13.3" (+2" = 15.3"). Too short.
  What about: 10 incs every 9 rows = 90 rows = 15" (+2" = 17")? 

Honestly, for a beginner pattern, I'll round things a bit:
- S: inc every 10 rows × 9 = 90 rows → 15" (+ 2" rib = 17" total sleeve)
- M: inc every 10 rows × 10 = 100 rows → 16.7" (+ 2" rib = 18.7" total sleeve) 
  → adjust: inc every 9 rows × 10 = 90 rows → 15" (+ 2" rib = 17" total). Then work 2 more inches even to get to 17" + 2" = 19"? No...
  
Let me just set all sleeve total lengths and compute from there:

Sleeve lengths (underarm to wrist, including 2" cuff):
- S: 17"
- M: 17.5"
- L: 18"
- XL: 18.5"
- 2XL: 19"

Rows for stockinette portion (after 2" rib, before cap):
- S: (17-2)" × 6 = 90 rows
- M: (17.5-2)" × 6 = 93 rows
- L: (18-2)" × 6 = 96 rows
- XL: (18.5-2)" × 6 = 99 rows
- 2XL: (19-2)" × 6 = 102 rows

Required increases:
- S: 9 increases over 90 rows → every 10 rows ✓ (9 × 10 = 90) ✓
- M: 10 increases over 93 rows → Let me use 9 increases every 9 rows (81 rows), then work even 12 rows. OR 10 × 9 = 90 rows, then 3 rows even. I'll say: work increase row, then work 8 rows even, repeat (i.e., every 9 rows) = worked at rows 9, 18, 27, 36, 45, 54, 63, 72, 81, 90 = 10 increases. Then 3 rows even. OK. But knitters often find "every Xth row" vs "every X rows" confusing. I'll state it as "every 9th row."

Wait, "every 10 rows" vs "every 10th row":
- "Every 10 rows" = work 9 rows, then inc on row 10, then 9 more, then inc... = increases at rows 10, 20, 30, etc.
- "Every 10th row" = same thing.

OK to just say "increase 1 st each end every 10th row X times" for beginners.

Let me settle on:
- S: every 10th row × 9 = 90 rows of sleeve, then 0 rows even before cap ✓
- M: every 9th row × 10 = 90 rows + 3 rows even = 93 rows ✓
- L: every 8th row × 12 = 96 rows ✓
- XL: every 8th row × 12 = 96 rows + 3 rows even = 99 rows ✓ (or say every 9th row × 11 = 99 ✓, but only 11 increases → 44+22 = 66... wait no.)

For XL: need 12 increases, 99 rows.
every 8th row × 12 = 96 rows, +3 even = 99 ✓. Simple enough.

For 2XL: 12 increases, 102 rows.
every 8th row × 12 = 96 rows + 6 even = 102 ✓

OK this works. Let me just note "work even until sleeve measures X" which is cleaner for beginners.

**SLEEVE CAP SHAPING**:
Sleeve cap height ≈ armhole depth - 1": 
- S/M: 6.5"
- L: 7"
- XL/2XL: 7.5"

Rows for cap:
- S/M: 6.5 × 6 = 39 rows
- L: 7 × 6 = 42 rows
- XL/2XL: 7.5 × 6 = 45 rows

For set-in sleeve cap: 
1. Bind off same as body underarm bind-off (same sts) at beg of next 2 rows
2. Dec 1 st each end every RS row until cap is about 2" wide
3. Bind off remaining sts

Let me work through S:
Sleeve at cap = 58 sts
Step 1: BO 4 sts each end (same as body) × 2 rows = 58 - 8 = 50 sts remaining
Step 2: Dec every RS row. Cap height = 39 rows total.
   - 2 rows used in step 1
   - 37 rows remaining = ~18 RS rows + 1 WS row... approximately
   - Need to end with ~9-10 sts for bind off (2" = 9 sts)
   - Starting sts: 50
   - Need to decrease from 50 to ~9 = 41 sts total to remove = ~20 decs each side... 
   - Wait: each dec row removes 2 sts (1 each end). So 41 ÷ 2 = 20.5 → need 21 dec rows ≈ 42 rows? But I only have 37 rows remaining. That's too many.

Let me reconsider cap shaping. Standard set-in sleeve cap formula:
1. Bind off sts at underarm (same as body BO) - 2 rows
2. Dec every RS row - multiple times
3. Dec every row (both RS and WS) near the end for more rapid decrease
4. Bind off remaining 1.5-2" worth of sts

Let me redo for S:
58 sts at start of cap
- BO 4 sts beg next 2 rows: 50 sts remain (2 rows used)
- Dec 1 each end every RS row × ? times
- Dec 1 each end every row × ? times  
- BO remaining

Total rows available = ~39 rows
After initial BO: 37 rows remain, about 19 RS rows

For the cap to have a nice shape, we want:
- ~2/3 of cap rows = regular every-RS-row decs
- ~1/3 of cap rows = faster decs (every row)
- End with ~1.5" (7 sts) for BO

If I dec every RS row for 12 rows (using 24 rows): 50 - 24 = 26 sts. Rows used = 24. Remaining = 37-24 = 13 rows.
Then dec every row for 7 rows (removing 14 sts): 26-14 = 12 sts. Total rows used = 24+7 = 31. Remaining = 37-31 = 6 rows... hmm complex.

For a BEGINNER pattern, let me simplify the cap:
- BO sts matching body underarm BO at start
- Dec every RS row until 10-12 sts remain
- BO all remaining sts

S (58 sts):
- BO 4 beg next 2 rows: 50 sts
- Dec 1 each end every RS row until 10 sts remain:
  Need to remove 40 sts = 20 dec rows × 2 rows each = 40 rows
  But I only have 37 rows remaining after the BOs!

I need to reconsider sleeve widths or cap heights. Let me make the sleeve narrower or the cap taller.

Actually, the issue is the sleeve at underarm is too wide for the cap. Let me recalculate:

For set-in sleeves, a good rule of thumb: 
Cap height ≈ 2/3 to 3/4 of armhole depth
Available rows for decreasing (after initial BO) ≈ (cap height in rows) - 2

Alternative: use standard set-in cap formula:
The cap must have enough rows to decrease from the sleeve width to about 1.5-2" wide.
Sts to decrease per side = (sleeve width at cap start - remaining sts) / 2

For simplicity with beginners, let me use the standard 3-step approach:
1. Match body underarm BO (2 rows)
2. Dec every RS row × n times
3. BO remaining sts (should be about 1-1.5" worth)

This means: remaining sts after all shaping = about 1.5" × 4.5 = ~7 sts, or 1" = ~5 sts.
Let's say BO last 10 sts.

For S: 58 sts
- BO 4 beg next 2 rows: 50 sts (2 rows)
- Need to get to 10 sts: need to remove 40 sts
- Each dec row removes 2 sts (1 each end)
- Need 20 dec RS rows = 40 total rows
- BUT total cap rows = ~39, minus 2 for BO = 37 rows available
- 20 RS rows in 37 rows = 20 RS + 20 WS = 40 rows? Only have 37. Off by 3 rows.

Problem: either the sleeve is too wide, or the cap needs to be taller. Let me make cap ht = 7" for all sizes (standard for worsted):
7" × 6 rows = 42 rows. After 2 BO rows: 40 rows = 20 RS rows of decs.
50 - (20 × 2) = 50 - 40 = 10 sts remaining → BO 10 sts ✓

For S: cap height = 7", BO 4 each end, dec every RS row × 20 = BO 10 remaining ✓

Let me check: armhole depth for S = 7.5", cap height = 7". That leaves 0.5" at the very bottom of the armhole where the sleeve isn't being eased in — this is fine, that's the actual underarm section.

For M: 64 sts
- BO 4 each end: 56 sts (2 rows)
- Cap height 7" → 42 rows → after 2 rows: 40 rows = 20 RS dec rows
- 56 - 40 = 16 sts remaining → BO 16 sts → 16/4.5 = 3.6" — a bit wide for cap BO
- Target: 10-12 sts for BO
- Need: 56 - 12 = 44 decs = 22 RS rows = 44 rows — too many for 40 available

Hmm. The M sleeve is too wide. Let me revisit sleeve widths.

Actually, set-in sleeve caps for worsted weight sweaters usually have a 3-step dec:
1. BO for underarm (same as body)
2. Dec every RS row × n1
3. Dec every row × n2 (faster decrease at top)
4. BO remaining

This is slightly more complex but necessary for a proper cap. For a "beginner" level, I can simplify by grouping steps 2 and 3, or by choosing a narrower sleeve.

Alternative: Let me use narrower sleeve widths at underarm:
- S: 12" → 54 sts
- M: 13" → 58 sts (rounds to 58, even ✓)  
- L: 14" → 63 → 62 sts (even ✓)
- XL: 15" → 67.5 → 68 sts
- 2XL: 16" → 72 sts

Hmm, let me try a different approach. Let me use the standard formula for set-in sleeve caps:

For the cap to work properly:
Total sts to eliminate per side = (sleeve sts at cap - BO sts) / 2
This is done in 3 phases:
Phase 1: BO same as underarm BO (already counted)
Phase 2: Dec 1 each end every RS row × n times
Phase 3: Dec 1 each end every row × m times  
Phase 4: BO remaining 10-12 sts

Let me just design the sleeves and caps to work out cleanly.

I'll work backwards from the desired cap height and end sts:

S: armhole depth = 7.5"
Cap height = 6.5" (armhole depth - 1" for ease of seaming)
Total cap rows = 6.5 × 6 = 39 rows

Sleeve at cap sts: Let's call it X.
BO 4 each side (2 rows): X - 8 sts remain, 37 rows left
Dec every RS row × 14 times (28 rows): X - 8 - 28 = remaining after phase 2, 9 rows left (= 4 RS rows + 5... actually 4 RS rows + 5 WS rows = 9 rows)
Then dec every row × 4 times if needed... 

This is getting too complicated to optimize cleanly. For a BEGINNER pattern with worsted weight, let me use a simplified approach that's commonly used in beginner patterns:

**Simplified sleeve cap for beginners**:
1. BO for underarm (same as body) at beg of next 2 rows
2. Dec 1 st each end every RS row until 1.5-2" remain
3. BO all remaining sts

And I'll size the sleeve widths so this works out to approximately the correct cap height.

For the math to work:
Rows needed for phase 2 = (sleeve sts - underarm BO × 2 - remaining sts) / 2 × 2
= (sleeve sts - underarm BO × 2 - remaining sts) rows total for phase 2
Plus 2 rows for phase 1.
Total = this + 2 should = cap height rows

Let me just define appropriate sleeve widths that work:

Cap height = armhole depth - 1":
- S/M: 6.5" = 39 rows
- L: 7" = 42 rows
- XL/2XL: 7.5" = 45 rows

End with ~10 sts (1 bind off row)
After phase 1 (2 rows): remaining rows = cap rows - 2 = phase 2 rows
Phase 2 RS rows = (remaining rows - 1) / 2 (round down) 
   Since we're decreasing every RS row, in X RS rows we use X RS + X WS = 2X rows...
   Wait: dec every RS row. That means: RS row (dec), WS row (no dec), RS row (dec), WS row...
   For N dec rows (each on RS), we use N × 2 = 2N rows.
   Then we have 1 more row (or 0) to BO.
   
   Cap rows after phase 1 = cap rows - 2
   Need: 2N rows (phase 2) + 1 row (BO) ≤ cap rows - 2
   N = (cap rows - 3) / 2

   Each dec row removes 2 sts (1 each end).
   Sts removed in phase 2 = 2N = cap rows - 3

   So: sleeve sts - (2 × underarm BO) - 2N = remaining sts for BO
   Sleeve sts = remaining sts + 2 × underarm BO + 2N
   Sleeve sts = 10 + 2 × BO + (cap rows - 3)

Let me calculate for each size:
- S: sleeve sts = 10 + 2(4) + (39-3) = 10 + 8 + 36 = 54 sts
- M: sleeve sts = 10 + 2(4) + (39-3) = 10 + 8 + 36 = 54 sts  
  (M has same cap height as S)
- L: sleeve sts = 10 + 2(5) + (42-3) = 10 + 10 + 39 = 59 → round to 60 (even)
- XL: sleeve sts = 10 + 2(5) + (45-3) = 10 + 10 + 42 = 62 sts
- 2XL: sleeve sts = 10 + 2(6) + (45-3) = 10 + 12 + 42 = 64 sts

Hmm, these seem a bit narrow. Let me recalculate with a larger end BO:

Use 14 sts for BO (about 3"):
- S: 14 + 8 + 36 = 58 sts → 58/4.5 = 12.9" ✓ (reasonable)
- M: 14 + 8 + 36 = 58 sts → same. Same size as S? That's odd.
- L: 14 + 10 + 39 = 63 → 64 sts → 64/4.5 = 14.2" ✓
- XL: 14 + 10 + 42 = 66 sts → 66/4.5 = 14.7" ✓
- 2XL: 14 + 12 + 42 = 68 sts → 68/4.5 = 15.1" ✓

But S and M same sleeve width is odd. M should be wider. Let me allow different cap end sts:
- S: end 14 sts, cap 39 rows → sleeve 58 sts → (39-3)/2 = 18 dec rows → removes 36 sts from 50 = 14 sts remaining ✓
- M: end 12 sts, cap 39 rows → sleeve = 12 + 8 + 36 = 56 sts → 56/4.5 = 12.4" 
  Hmm, M should be wider than S. 

I think I need to adjust the cap height for M or allow M to have more rows available.

Let me try: give M slightly more cap height: 7" = 42 rows
- M: end 12 sts, cap 42 rows → sleeve = 12 + 8 + (42-3) = 12 + 8 + 39 = 59 → 60 sts → 60/4.5 = 13.3" ✓

OK so:
- S: cap 6.5" (39 rows), end 14 sts, sleeve = 14 + 8 + 36 = 58 sts
- M: cap 7" (42 rows), end 12 sts, sleeve = 12 + 8 + 39 = 59 → 60 sts
- L: cap 7.5" (45 rows), end 12 sts, sleeve = 12 + 10 + 42 = 64 sts
- XL: cap 7.5" (45 rows), end 10 sts, sleeve = 10 + 10 + 42 = 62 sts
- 2XL: cap 7.5" (45 rows), end 10 sts, sleeve = 10 + 12 + 42 = 64 sts

Let me verify these widths are reasonable:
- S: 58 sts = 58/4.5 = 12.9" ≈ 13" ✓
- M: 60 sts = 60/4.5 = 13.3" ≈ 13.5" ✓
- L: 64 sts = 64/4.5 = 14.2" ≈ 14" ✓  
- XL: 62 sts = 62/4.5 = 13.8" ≈ 14" ✓ (should be wider than L for XL)... 
  Hmm, XL should be wider than L. Let me swap: L=62, XL=64? Or give XL 14 sts end BO:
  XL: end 14 sts, sleeve = 14 + 10 + 42 = 66 sts → 66/4.5 = 14.7" ✓
- 2XL: end 10 sts, sleeve = 10 + 12 + 42 = 64 sts... same as L. Not right.
  2XL: end 12 sts, sleeve = 12 + 12 + 42 = 66 sts → same as XL. Close but ok.
  OR: 2XL: end 10 sts, longer cap: cap 8" (48 rows), sleeve = 10 + 12 + (48-3) = 10 + 12 + 45 = 67 → 68 sts

This is getting really messy. Let me step back and use a more practical approach.

For a beginner pattern, many designers use a simplified "set-in" that still looks proper but uses a cleaner formula. Let me use this approach:

**PRACTICAL SLEEVE CAP FORMULA (Simplified)**:

Given sleeve sts at underarm (X), and armhole depth (D):
1. BO: same as body underarm BO, beg of next 2 rows
2. Dec 1 ea end EOR (every other row = every RS row):
   - Number of dec rows = approximately (X/2 - 5) to (X/2 - 7)  
   - This leaves about 10-14 sts
3. BO all remaining sts

This doesn't perfectly match the cap height to the armhole, but it works for a beginner pattern. The slight mismatch gets eased in during seaming.

Let me just use proportional sleeve widths and calculate naturally:

I'll define:
- Sleeve underarm sts (before cap) based on reasonable arm width measurements
- Calculate cap rows needed
- See if cap height ≈ armhole depth - 1"

Sleeve underarm widths:
- S: 13" → 58 sts (even ✓) 
- M: 14" → 63 → 62 sts (even ✓)
- L: 15" → 67.5 → 68 sts (even ✓)
- XL: 16" → 72 sts (even ✓)
- 2XL: 17" → 76.5 → 76 sts (even ✓)

Cap shaping:
1. BO same as body underarm: each side (2 rows)
After BO:
- S: 58 - 8 = 50 sts; then dec every RS row: (50-14)/2 = 18 RS rows = 36 rows + BO row + 2 for initial BO = 39 rows. Cap ht = 39/6 = 6.5" ✓ (armhole 7.5", cap 6.5", diff = 1" ✓)
- M: 62 - 8 = 54 sts; dec every RS: (54-12)/2 = 21 RS rows = 42 rows + 1 BO + 2 BO = 45 rows. Cap ht = 45/6 = 7.5". Armhole = 7.5". Cap ht should be < armhole... maybe that's OK, cap = armhole depth - 0" means the cap exactly fills the armhole. Standard says 0-1" less. 7.5" cap for 7.5" armhole is tight but possible.
   Let me adjust: (54-16)/2 = 19 dec rows = 38 rows + 2+1 = 41 rows = 6.8". Better.
   So BO 16 sts at end for M.
   
- L: 68 - 10 = 58 sts; dec every RS: (58-12)/2 = 23 dec rows = 46 rows + 2+1 = 49 rows. Cap ht = 49/6 = 8.2". Armhole = 8". Cap > armhole — problem!
   Need fewer decs. (58-20)/2 = 19 dec rows = 38 + 3 = 41 rows = 6.8". End BO = 20 sts = 4.4". Too wide to BO.
   Try: (58-14)/2 = 22 dec rows = 44 rows = 7.3" + 2 = 7.3". Armhole 8", cap 7.3". Diff = 0.7". OK.
   End with 14 sts, add 1 BO row: 44 + 2 + 1 = 47 rows. 47/6 = 7.8". Hmm.
   
This is still complex. Let me just use the 3-step approach and make it clean for the beginner:

**Final simplified decision**: 

I'll include a 3-row approach:
1. BO for underarm (2 rows)
2. Dec 1 ea end every RS row × N times
3. BO remaining sts

I'll pick N for each size to give reasonable cap heights, and note the final BO stitch count.

Let me define sleeve sts, then pick N, calculate end sts and cap ht:

- S: 58 sts, BO 4×2=8, start with 50. Dec every RS ×18=36 dec sts, end with 50-36=14 sts. Total rows = 2 + 36 + 1 = 39 rows = 6.5". Armhole 7.5". ✓ (1" difference, good)
- M: 62 sts, BO 4×2=8, start with 54. Dec every RS ×18=36, end with 18 sts. Total = 2 + 36 + 1 = 39 rows = 6.5". Armhole 7.5" ✓ (1" diff)
- L: 68 sts, BO 5×2=10, start with 58. Dec every RS ×20=40, end with 18 sts. Total = 2 + 40 + 1 = 43 rows = 7.2". Armhole 8". ✓ (0.8" diff)
- XL: 72 sts, BO 5×2=10, start with 62. Dec every RS ×22=44, end with 18 sts. Total = 2 + 44 + 1 = 47 rows = 7.8". Armhole 8.5" ✓ (0.7" diff)
- 2XL: 76 sts, BO 6×2=12, start with 64. Dec every RS ×22=44, end with 20 sts. Total = 2 + 44 + 1 = 47 rows = 7.8". Armhole 8.5" ✓ (0.7" diff)

Good! Let me verify:
- S: 50 - 36 = 14 sts BO at end. 14/4.5 = 3.1" ✓ (reasonable to BO)
- M: 54 - 36 = 18 sts BO. 18/4.5 = 4" ✓ 
- L: 58 - 40 = 18 sts BO. 4" ✓
- XL: 62 - 44 = 18 sts BO. 4" ✓
- 2XL: 64 -