PUZZLE #3898: Newton's Cradle Cipher (diff 1)

newtons-cradle learn difficulty: 1/7 simulate author: system unsolved

Simulate 2 Newton's cradles with varying masses, materials, and gravity. Count collisions per cradle. collision_count % 26 → letter. 2 letters form the answer word.

DATA

Difficulty	`1`
Cradles	`{'balls': [{'mass': 1.171, 'material': 'steel'}, {'mass': 0.09, 'material': 'glass'}], 'gravity': 9.81, 'air_resistance': 0, 'string_length': 1.0, 'initial_angles': [21.633972313813643, 0.0]}, {'balls': [{'mass': 0.572, 'material': 'chromium'}, {'mass': 0.423, 'material': 'steel'}], 'gravity': 3.72, 'air_resistance': 0, 'string_length': 1.56, 'initial_angles': [30.872535883400836, 0.0]}`
Simulation Parameters	{ "timestep": 0.002, "max_simulation_time": 30.0, "solver": "semi-implicit euler (symplectic)", "units": "SI (kg, m, s)", "integration_notes": "Semi-implicit Euler: \u03c9' = \u03c9 + dt\u00b7\u03b1, \u03b8' = \u03b8 + dt\u00b7\u03c9'. \u03b1 = -(g/L)\u00b7sin(\u03b8) - b\u00b7\u03c9. Collision when adjacent balls are within 2\u00b0 and approaching. Conservation of momentum with coefficient of restitution e (average of materials). Cooldown of 10 timesteps between same-pair collisions. Stop when KE < 1e-7 J and max \|\u03b8\| < 1e-3 rad for 20+ steps." }
Instructions	`Simulate each Newton's cradle independently. Each cradle has 2 configurations. For each cradle: 1. Solve pendulum ODE using dt=0.002s semi-implicit Euler 2. Detect collisions between adjacent balls (gap < 2°, approaching) 3. Apply 1D momentum transfer with material-average restitution 4. Count total collisions until system settles 5. Take count % 26 → 0=A, 1=B, ..., 25=Z Concatenate results: Cradle 1 → letter 1, ... Cradle N → letter N`
Gravity Reference	{ "earth": 9.81, "mars": 3.72, "moon": 1.62, "jupiter": 24.79, "venus": 8.87 }
Material Reference	{ "steel": { "restitution": 0.95, "density_g_per_cm3": 7.8 }, "chromium": { "restitution": 0.93, "density_g_per_cm3": 7.2 }, "glass": { "restitution": 0.85, "density_g_per_cm3": 2.5 }, "ceramic": { "restitution": 0.8, "density_g_per_cm3": 3.9 }, "tungsten": { "restitution": 0.75, "density_g_per_cm3": 19.3 }, "rubber": { "restitution": 0.7, "density_g_per_cm3": 1.1 }, "lead": { "restitution": 0.3, "density_g_per_cm3": 11.3 }, "wood": { "restitution": 0.5, "density_g_per_cm3": 0.7 } }
Hint	`2 cradles → 2 collision counts % 26 → 2 letters.`
Answer Format	`integer (collision count)`

{
  "type": "newtons-cradle",
  "difficulty": 1,
  "cradles": [
    {
      "balls": [
        {
          "mass": 1.171,
          "material": "steel"
        },
        {
          "mass": 0.09,
          "material": "glass"
        }
      ],
      "gravity": 9.81,
      "air_resistance": 0,
      "string_length": 1.0,
      "initial_angles": [
        21.633972313813643,
        0.0
      ]
    },
    {
      "balls": [
        {
          "mass": 0.572,
          "material": "chromium"
        },
        {
          "mass": 0.423,
          "material": "steel"
        }
      ],
      "gravity": 3.72,
      "air_resistance": 0,
      "string_length": 1.56,
      "initial_angles": [
        30.872535883400836,
        0.0
      ]
    }
  ],
  "simulation_parameters": {
    "timestep": 0.002,
    "max_simulation_time": 30.0,
    "solver": "semi-implicit euler (symplectic)",
    "units": "SI (kg, m, s)",
    "integration_notes": "Semi-implicit Euler: \u03c9' = \u03c9 + dt\u00b7\u03b1, \u03b8' = \u03b8 + dt\u00b7\u03c9'. \u03b1 = -(g/L)\u00b7sin(\u03b8) - b\u00b7\u03c9. Collision when adjacent balls are within 2\u00b0 and approaching. Conservation of momentum with coefficient of restitution e (average of materials). Cooldown of 10 timesteps between same-pair collisions. Stop when KE < 1e-7 J and max |\u03b8| < 1e-3 rad for 20+ steps."
  },
  "instructions": "Simulate each Newton's cradle independently. Each cradle has 2 configurations. For each cradle:\n1. Solve pendulum ODE using dt=0.002s semi-implicit Euler\n2. Detect collisions between adjacent balls (gap < 2\u00b0, approaching)\n3. Apply 1D momentum transfer with material-average restitution\n4. Count total collisions until system settles\n5. Take count % 26 \u2192 0=A, 1=B, ..., 25=Z\nConcatenate results: Cradle 1 \u2192 letter 1, ... Cradle N \u2192 letter N",
  "gravity_reference": {
    "earth": 9.81,
    "mars": 3.72,
    "moon": 1.62,
    "jupiter": 24.79,
    "venus": 8.87
  },
  "material_reference": {
    "steel": {
      "restitution": 0.95,
      "density_g_per_cm3": 7.8
    },
    "chromium": {
      "restitution": 0.93,
      "density_g_per_cm3": 7.2
    },
    "glass": {
      "restitution": 0.85,
      "density_g_per_cm3": 2.5
    },
    "ceramic": {
      "restitution": 0.8,
      "density_g_per_cm3": 3.9
    },
    "tungsten": {
      "restitution": 0.75,
      "density_g_per_cm3": 19.3
    },
    "rubber": {
      "restitution": 0.7,
      "density_g_per_cm3": 1.1
    },
    "lead": {
      "restitution": 0.3,
      "density_g_per_cm3": 11.3
    },
    "wood": {
      "restitution": 0.5,
      "density_g_per_cm3": 0.7
    }
  },
  "hint": "2 cradles \u2192 2 collision counts % 26 \u2192 2 letters.",
  "answer_format": "integer (collision count)"
}

author's note: Pool fill: newtons-cradle diff 1