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whong92/jane-street-nov.ipynb

Last active November 26, 2024 22:06
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jane-street-nov.ipynb
{
"cells": [
{
"cell_type": "markdown",
"id": "be5abb8b-288f-48d6-a04c-87436e23e686",
"metadata": {
"execution": {
"iopub.execute_input": "2024-11-26T21:17:03.941015Z",
"iopub.status.busy": "2024-11-26T21:17:03.940687Z",
"iopub.status.idle": "2024-11-26T21:17:03.944196Z",
"shell.execute_reply": "2024-11-26T21:17:03.943632Z",
"shell.execute_reply.started": "2024-11-26T21:17:03.940994Z"
},
"tags": []
},
"source": [
"# Jane Street Nov 2024 Puzzle Solution"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "b9254ed3-6713-48c3-9895-9ea29d2a9a60",
"metadata": {},
"outputs": [],
"source": [
"import numpy as np\n",
"import matplotlib.pyplot as plt\n",
"from scipy.integrate import nquad, tplquad, dblquad\n",
"from typing import Any"
]
},
{
"cell_type": "code",
"execution_count": 6,
"id": "cdfeb7d4-86c4-4d89-bd14-4bda8dd283a1",
"metadata": {
"execution": {
"iopub.execute_input": "2024-11-26T21:08:46.161643Z",
"iopub.status.busy": "2024-11-26T21:08:46.161183Z",
"iopub.status.idle": "2024-11-26T21:08:46.164792Z",
"shell.execute_reply": "2024-11-26T21:08:46.164338Z",
"shell.execute_reply.started": "2024-11-26T21:08:46.161620Z"
},
"tags": []
},
"outputs": [],
"source": [
"xyb = np.array([[0.01, 0.05]])\n",
"xyr = np.array([[0.8, 0.1]])"
]
},
{
"cell_type": "code",
"execution_count": 7,
"id": "da283f9d-7b5c-40c6-b877-481b1c8c4b3c",
"metadata": {
"execution": {
"iopub.execute_input": "2024-11-26T21:08:46.165864Z",
"iopub.status.busy": "2024-11-26T21:08:46.165583Z",
"iopub.status.idle": "2024-11-26T21:08:46.171958Z",
"shell.execute_reply": "2024-11-26T21:08:46.171519Z",
"shell.execute_reply.started": "2024-11-26T21:08:46.165836Z"
},
"tags": []
},
"outputs": [],
"source": [
"def get_xc(xyb, xyr):\n",
" \"\"\"get the x coordinate of the intersection point.\"\"\"\n",
" xb = xyb[:, 0]\n",
" xr = xyr[:, 0]\n",
" return (-np.power(np.linalg.norm(xyb, axis=1), 2) + np.power(np.linalg.norm(xyr, axis=1), 2)) / 2 / (xr - xb)"
]
},
{
"cell_type": "code",
"execution_count": 8,
"id": "9253a994-02d4-49b3-a4fd-2301df113d10",
"metadata": {
"execution": {
"iopub.execute_input": "2024-11-26T21:08:46.835603Z",
"iopub.status.busy": "2024-11-26T21:08:46.835269Z",
"iopub.status.idle": "2024-11-26T21:08:46.838645Z",
"shell.execute_reply": "2024-11-26T21:08:46.838100Z",
"shell.execute_reply.started": "2024-11-26T21:08:46.835580Z"
},
"tags": []
},
"outputs": [],
"source": [
"xc = get_xc(xyb, xyr)"
]
},
{
"cell_type": "code",
"execution_count": 9,
"id": "aa4fec70-2dba-41af-a62e-d96df6bef818",
"metadata": {
"execution": {
"iopub.execute_input": "2024-11-26T21:08:47.453249Z",
"iopub.status.busy": "2024-11-26T21:08:47.452928Z",
"iopub.status.idle": "2024-11-26T21:08:47.457051Z",
"shell.execute_reply": "2024-11-26T21:08:47.456520Z",
"shell.execute_reply.started": "2024-11-26T21:08:47.453228Z"
},
"tags": []
},
"outputs": [],
"source": [
"def get_xc_ind(yb, xb, xr, yr):\n",
" \"\"\"returns boolean indicator for whether xc is less than 1 and more than 0\"\"\"\n",
" diag_indicator = yb > xb\n",
" yb_orig = yb.copy()\n",
" xb_orig = xb.copy()\n",
" yb[diag_indicator] = xb_orig[diag_indicator]\n",
" xb[diag_indicator] = yb_orig[diag_indicator]\n",
" xc = get_xc(\n",
" np.stack((xb, yb), axis=1),\n",
" np.stack((xr, yr), axis=1)\n",
" )\n",
" return np.logical_and((0. <= xc), (xc <= 1.)).astype(np.float64)"
]
},
{
"cell_type": "code",
"execution_count": 47,
"id": "c0ee8495-3f69-4f8d-9de6-069dd622fc11",
"metadata": {
"execution": {
"iopub.execute_input": "2024-11-26T22:03:48.373803Z",
"iopub.status.busy": "2024-11-26T22:03:48.373310Z",
"iopub.status.idle": "2024-11-26T22:03:48.378143Z",
"shell.execute_reply": "2024-11-26T22:03:48.377678Z",
"shell.execute_reply.started": "2024-11-26T22:03:48.373771Z"
},
"tags": []
},
"outputs": [],
"source": [
"def visualize_mc(xb: float, yb: float, num_samples: int):\n",
" \"\"\"Visualize MC simulation for a single xb, yb\"\"\"\n",
" xr = np.random.uniform(0., 1.0, num_samples)\n",
" yr = np.random.uniform(0., 1.0, num_samples)\n",
" xb = xb * np.ones(shape=(num_samples))\n",
" yb = yb * np.ones(shape=(num_samples))\n",
" xc_ind = get_xc_ind(yb, xb, xr, yr).astype(np.bool)\n",
" xr_valid = xr[xc_ind]\n",
" yr_valid = yr[xc_ind]\n",
" print(\"MC Probability = \", np.mean(xc_ind))\n",
" plt.scatter(xr_valid, yr_valid, marker='.')\n",
" plt.ylim([0., 1.])\n",
" plt.xlim([0., 1.])\n",
" plt.show()"
]
},
{
"cell_type": "code",
"execution_count": 48,
"id": "ce0e4b92-3594-47d8-b2c6-598dafc21338",
"metadata": {
"execution": {
"iopub.execute_input": "2024-11-26T22:03:48.685499Z",
"iopub.status.busy": "2024-11-26T22:03:48.685172Z",
"iopub.status.idle": "2024-11-26T22:03:48.689295Z",
"shell.execute_reply": "2024-11-26T22:03:48.688785Z",
"shell.execute_reply.started": "2024-11-26T22:03:48.685478Z"
},
"tags": []
},
"outputs": [],
"source": [
"def get_naive_mc(num_samples):\n",
" \"\"\"Calculates naive MC estimate of the probability.\"\"\"\n",
" xb = np.random.uniform(0., 0.5, num_samples)\n",
" yb = np.random.uniform(0., 0.5, num_samples)\n",
" xr = np.random.uniform(0., 1.0, num_samples)\n",
" yr = np.random.uniform(0., 1.0, num_samples)\n",
" xc_ind = get_xc_ind(yb, xb, xr, yr)\n",
" return np.mean(xc_ind)"
]
},
{
"cell_type": "code",
"execution_count": 45,
"id": "8842c973-7ae9-42ee-8d96-a49f92194023",
"metadata": {
"execution": {
"iopub.execute_input": "2024-11-26T22:03:19.619520Z",
"iopub.status.busy": "2024-11-26T22:03:19.619211Z",
"iopub.status.idle": "2024-11-26T22:03:21.209067Z",
"shell.execute_reply": "2024-11-26T22:03:21.208482Z",
"shell.execute_reply.started": "2024-11-26T22:03:19.619502Z"
},
"tags": []
},
"outputs": [
{
"data": {
"text/plain": [
"np.float64(0.4916135)"
]
},
"execution_count": 45,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"get_naive_mc(10000000)"
]
},
{
"cell_type": "code",
"execution_count": 49,
"id": "99dfa7c9-d134-46cd-abc8-924e5082e7ec",
"metadata": {
"execution": {
"iopub.execute_input": "2024-11-26T22:03:50.362200Z",
"iopub.status.busy": "2024-11-26T22:03:50.361882Z",
"iopub.status.idle": "2024-11-26T22:03:50.492529Z",
"shell.execute_reply": "2024-11-26T22:03:50.491959Z",
"shell.execute_reply.started": "2024-11-26T22:03:50.362182Z"
},
"tags": []
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"MC Probability = 0.47767\n"
]
},
{
"data": {
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\n",
"text/plain": [
"<Figure size 640x480 with 1 Axes>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"visualize_mc(0.4, 0.3, 100000)"
]
},
{
"cell_type": "code",
"execution_count": 50,
"id": "ff7a1676-fd7e-4ef0-8264-0bbb3df8edb1",
"metadata": {
"execution": {
"iopub.execute_input": "2024-11-26T22:03:57.514270Z",
"iopub.status.busy": "2024-11-26T22:03:57.513836Z",
"iopub.status.idle": "2024-11-26T22:03:57.518379Z",
"shell.execute_reply": "2024-11-26T22:03:57.517862Z",
"shell.execute_reply.started": "2024-11-26T22:03:57.514251Z"
},
"tags": []
},
"outputs": [],
"source": [
"def circles_sum(yb, xb):\n",
" \"\"\"Sum of the above circles areas. Union plus intersection.\"\"\"\n",
" return 0.25 * np.pi * (\n",
" np.power(xb, 2.) + \n",
" 2 * np.power(yb, 2.) + \n",
" np.power(1. - xb, 2.)\n",
" )\n",
"\n",
"def circles_intersection(yb: float, xb: float) -> Any:\n",
" \"\"\"The intersection to be subtracted.\"\"\"\n",
" return 0.5 * (\n",
" np.arctan2(yb, xb) * (np.power(xb, 2.) + np.power(yb, 2.)) + \n",
" np.arctan2(yb, 1 - xb) * (np.power(1. - xb, 2.) + np.power(yb, 2.)) -\n",
" yb\n",
" )"
]
},
{
"cell_type": "code",
"execution_count": 51,
"id": "8aed9725-01fc-4a3f-b27d-a7491da6f4ad",
"metadata": {
"execution": {
"iopub.execute_input": "2024-11-26T22:03:57.681965Z",
"iopub.status.busy": "2024-11-26T22:03:57.681644Z",
"iopub.status.idle": "2024-11-26T22:03:57.686108Z",
"shell.execute_reply": "2024-11-26T22:03:57.685586Z",
"shell.execute_reply.started": "2024-11-26T22:03:57.681944Z"
},
"tags": []
},
"outputs": [
{
"data": {
"text/plain": [
"np.float64(0.48026201312952993)"
]
},
"execution_count": 51,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# calculate for 1 point\n",
"(circles_sum(0.3, 0.4) - 2 * circles_intersection(0.3, 0.4))"
]
},
{
"cell_type": "code",
"execution_count": 52,
"id": "8dd24cba-49ae-4183-9b54-146c758c9119",
"metadata": {
"execution": {
"iopub.execute_input": "2024-11-26T22:03:58.281630Z",
"iopub.status.busy": "2024-11-26T22:03:58.281317Z",
"iopub.status.idle": "2024-11-26T22:03:58.289980Z",
"shell.execute_reply": "2024-11-26T22:03:58.289441Z",
"shell.execute_reply.started": "2024-11-26T22:03:58.281608Z"
},
"tags": []
},
"outputs": [],
"source": [
"# integrate\n",
"integral_circles_union = dblquad(circles_sum, 0, 0.5, 0., lambda x: x)\n",
"integral_circles_intersection = dblquad(circles_intersection, 0, 0.5, 0., lambda x: x)\n",
"area = 8.0 * (integral_circles_union[0] - 2 * integral_circles_intersection[0])"
]
},
{
"cell_type": "code",
"execution_count": 53,
"id": "2a4d962e-ea60-431c-afe4-f1814fdf8d25",
"metadata": {
"execution": {
"iopub.execute_input": "2024-11-26T22:03:58.910108Z",
"iopub.status.busy": "2024-11-26T22:03:58.909791Z",
"iopub.status.idle": "2024-11-26T22:03:58.913766Z",
"shell.execute_reply": "2024-11-26T22:03:58.913248Z",
"shell.execute_reply.started": "2024-11-26T22:03:58.910089Z"
},
"tags": []
},
"outputs": [
{
"data": {
"text/plain": [
"0.49140757883830805"
]
},
"execution_count": 53,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"area"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "236ddbbb-c50e-4ecf-8427-bd845db8688d",
"metadata": {},
"outputs": [],
"source": []
}
],
"metadata": {
"kernelspec": {
"display_name": "Python 3 (ipykernel)",
"language": "python",
"name": "python3"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.10.13"
}
},
"nbformat": 4,
"nbformat_minor": 5
}
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