Motivation ========== I just wanted to combine arrows and thermos intimately, and the saxophone idea dropped out almost immediately. Being able to find almost identical-posed players at different positions relative to box boundaries that admitted a solution was a bonus. I added all three notes for harmony - two of them aren't even necessary! Solve Overview ============== - Pencil the players - Near-solve the easy player - Grind down the bounds on the others bit by bit - Fill in the gaps using sudoku - Recognise the interdependencies, which admit a non-standard colouring - Find the cracks - Speedrun Pencil the Players ------------------ The first thing to notice is where the two arms can't both be the minimal possible values simultaniously. E.g. the bass sax can't have a 6 on his right hand (on the left, he's facing us), because that requires a [12] up the right arm and a 1 up the left arm. Other right hands and thus horns are similarly restricted below what a thermo would normally require, and likewise their heads will be higher - heads below 7 is an absurdity for the rightmost 3, but we can do better. Near-solve the Easy Player -------------------------- The chappy on the right has is whole torso in one box, so there can be no repeated values. You simply can't increment and sum values less than 9 for the head, with [13] and 2 along the arms, and 4 on the horn. Grind Down Bounds ----------------- The same arm logic pushes bass dude's head up to 9, likewise bottom right guy. Locations of 8s and 9s in boxes are quite restricted, because they can't be on arms or down the sax (except for the player's head). This prevents bottom left guy from having 9 on his head. Almost all the arms can be wittled down to being from [123] quite quickly, 4s are just too expensive. Likewise, you'll not get any head below 8, and in fact four have head=9. Due to alignment, most of the 1s, 2s, and 3s will place themselves on the arms. Their left hands (on our right) will become 7s and 8s, their right hands will be limited to at most two values. Fill in Gaps Using Sudoku ------------------------- At this point, a quick scan will resolve many more cells to unique values or pairs, with a few triplets dotted around. All of column 7, and box 6 should be full by now. Recognise Interdependencies and Colour -------------------------------------- There should be [23] and [56] pairs all over the place, and several of these are connected by sharing visibility with [25] and [36] cells. This admits a high-low colouring: If the left dude's upper arm is a low 2, his left hand's a high 6, r4c4 becomes low 5, r6c5 becomes low 2, r5c5 becomes high 3, r5c6 becomes high 6 (plus more in boxes 2 and 8). If the left dude's upper arm is a high 3, his left hand's a low 5, r4c4 becomes high 6, r5c6 bcomes low 3, r5c5 becomes low 2, r6c5 becomes high 5 (plus more in boxes 2 and 8). These two are hi-lo opposites of each other. Find the Cracks --------------- Bottom left dude doesn't succumb to the high/low colouring, but he does better than that. If left dude's upper arm is low, so is r6c5 (namely a 2), but if his upper arm is high, with r6c5 r6c1 is forced to be a 2, and either way so r6c2 is 3. Nearly there... Box 1 and column 3 are both missing their 5 presently, restricted to 2 cells each. If the intersection (r2c3) isn't the 5, the [36] pair formed puts a 5, a 2 and a [25] in column 1. => r2c6 is the 5. And that's huge. Speedrun -------- Everything just fills itself out at this point. Hopefully the little present in the top right reveals itself fairly late on in the process. Proof of uniqueness =================== Puzzle with 3 notes: https://sigh.github.io/Interactive-Sudoku-Solver/?q=.Thermo%7ER5C2%7ER4C2%7ER3C2%7ER2C2.Thermo%7ER5C5%7ER4C5%7ER3C5%7ER2C5%7ER1C5.Thermo%7ER6C8%7ER5C8%7ER4C8%7ER3C8.Thermo%7ER9C6%7ER8C6%7ER7C6%7ER6C6.Thermo%7ER8C3%7ER7C3%7ER6C3%7ER5C3.Arrow%7ER2C2%7ER3C3%7ER3C2.Arrow%7ER1C5%7ER2C6%7ER2C5.Arrow%7ER6C6%7ER7C7%7ER7C6.Arrow%7ER3C8%7ER4C7%7ER5C7%7ER5C8.Arrow%7ER3C8%7ER4C9%7ER4C8.Arrow%7ER1C5%7ER2C4%7ER3C4%7ER4C5.Arrow%7ER5C3%7ER6C4%7ER6C3.Arrow%7ER5C3%7ER6C2%7ER7C2%7ER7C3.Arrow%7ER6C6%7ER7C5%7ER8C5%7ER8C6.Arrow%7ER2C2%7ER3C1%7ER4C1%7ER4C2.BlackDot%7ER9C6%7ER9C7.BlackDot%7ER6C8%7ER6C9.BlackDot%7ER8C3%7ER8C4 Puzzle with only leftmost note: https://sigh.github.io/Interactive-Sudoku-Solver/?q=.Thermo%7ER5C2%7ER4C2%7ER3C2%7ER2C2.Thermo%7ER5C5%7ER4C5%7ER3C5%7ER2C5%7ER1C5.Thermo%7ER6C8%7ER5C8%7ER4C8%7ER3C8.Thermo%7ER9C6%7ER8C6%7ER7C6%7ER6C6.Thermo%7ER8C3%7ER7C3%7ER6C3%7ER5C3.Arrow%7ER2C2%7ER3C3%7ER3C2.Arrow%7ER1C5%7ER2C6%7ER2C5.Arrow%7ER6C6%7ER7C7%7ER7C6.Arrow%7ER3C8%7ER4C7%7ER5C7%7ER5C8.Arrow%7ER3C8%7ER4C9%7ER4C8.Arrow%7ER1C5%7ER2C4%7ER3C4%7ER4C5.Arrow%7ER5C3%7ER6C4%7ER6C3.Arrow%7ER5C3%7ER6C2%7ER7C2%7ER7C3.Arrow%7ER6C6%7ER7C5%7ER8C5%7ER8C6.Arrow%7ER2C2%7ER3C1%7ER4C1%7ER4C2.BlackDot%7ER8C3%7ER8C4 Puzzle with only middle note: https://sigh.github.io/Interactive-Sudoku-Solver/?q=.Thermo%7ER5C2%7ER4C2%7ER3C2%7ER2C2.Thermo%7ER5C5%7ER4C5%7ER3C5%7ER2C5%7ER1C5.Thermo%7ER6C8%7ER5C8%7ER4C8%7ER3C8.Thermo%7ER9C6%7ER8C6%7ER7C6%7ER6C6.Thermo%7ER8C3%7ER7C3%7ER6C3%7ER5C3.Arrow%7ER2C2%7ER3C3%7ER3C2.Arrow%7ER1C5%7ER2C6%7ER2C5.Arrow%7ER6C6%7ER7C7%7ER7C6.Arrow%7ER3C8%7ER4C7%7ER5C7%7ER5C8.Arrow%7ER3C8%7ER4C9%7ER4C8.Arrow%7ER1C5%7ER2C4%7ER3C4%7ER4C5.Arrow%7ER5C3%7ER6C4%7ER6C3.Arrow%7ER5C3%7ER6C2%7ER7C2%7ER7C3.Arrow%7ER6C6%7ER7C5%7ER8C5%7ER8C6.Arrow%7ER2C2%7ER3C1%7ER4C1%7ER4C2.BlackDot%7ER9C6%7ER9C7