yoniLavi · September 4, 2024 00:31
diff --git a/ballmer_binary_search_game.ipynb b/ballmer_binary_search_game.ipynb
 {
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This is a simulation of Steve Ballmer's binary search game, as described in [John Graham-Cumming's blog](https://blog.jgc.org/2024/09/steve-ballmers-binary-search-interview.html)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "Expected value per game: $0.20\n"
     ]
    }
   ],
   "source": [
    "from statistics import mean\n",
    "\n",
    "INITIAL_PAYMENT = 5  # dollars paid if first guess is correct, minus 1 for each incorrect guess\n",
    "MIN_GUESS = 1 # The range of numbers to guess from\n",
    "MAX_GUESS = 100\n",
    "\n",
    "def format_payout(payout: float, precision: int = 0) -> str:\n",
    "    return ('-' if payout < 0 else '') + f'${abs(payout):.{precision}f}'\n",
    "\n",
    "def calculate_payouts(low: int, high: int, depth: int = 0) -> list[float]:\n",
    "    if low > high:\n",
    "        return []\n",
    "\n",
    "    mid = (low + high) // 2\n",
    "    payout = INITIAL_PAYMENT - depth\n",
    "    left_payouts = calculate_payouts(low, mid - 1, depth + 1)\n",
    "    right_payouts = calculate_payouts(mid + 1, high, depth + 1)\n",
    "\n",
    "    return left_payouts + [payout] + right_payouts\n",
    "\n",
    "payouts = calculate_payouts(MIN_GUESS, MAX_GUESS)\n",
    "average_payout = mean(payouts)\n",
    "print(f\"\\nExpected value per game: {format_payout(average_payout, 2)}\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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eeUaGYbhtv/BXufY0//zzzwts2XAxitv2Dh06qGnTpkU+fnh4uJYvX67ly5fr66+/1qRJk5Samqpu3brp2LFjF3x806ZNddNNN5m3g4KC1KhRI+3bt88s8/LyMreQyM3NVXJysrnNTf7+dOnbt2+BLRruu+8+eXt7a968eWbZsmXLlJSUdMGV1iEhIdq+fbsGDRqklJQUTZs2TQ8++KCCg4M1duxYc+zExcVpz549evDBB5WcnKykpCQlJSUpPT1dXbp00ffff2/2af72nTlzRsnJyapfv778/f0Ljemxxx5z21N++fLlOnnypJ577rkC28nYbDa325UqVXKL0dPTU23atHF7jQuTlpZmPj6/pUuXKigoyPxXu3Ztt/uLcu65dOnSxe3bDm3btpUk9ejRw221rqvc1ebivNbn06dPH7fn6dmzp2rUqKGlS5eW2PNcd911qlSpkr7//ntJeSvOQ0ND1adPH/3444/KyMiQYRhat26d27mwaNEi3XTTTapatar5vElJSeratatyc3PN4y1atEhVqlTRLbfc4lYvIiJClSpVKrBdSlHOudLgcDjM3w9wOp06fvy4cnJy1KpVq0LHxv3332/uMy7JbLOrnYcPH1ZcXJz69u2rKlWqmPVuueWWYl3DiiIzM9Pytw+8vb2VmZlp1pNU6A8ou85TVx0AAFAy2M4FAADgPKZMmaKGDRvKw8NDISEhatSokdtevV999ZX++c9/Ki4uTllZWWb5uQnGPn366KOPPtLatWt1880367vvvtORI0f0yCOPmHV+//131axZs8BX8Js0aWLeX1Luv/9+TZ8+XY8++qiee+45denSRffee6969ux5UXsRF7ftderUKdbxfX191bVrV/P2bbfdphtvvFGtWrXSv/71L7355pvnfXxYWFiBsqpVq7rt5ex0OjV58mRNnTpV8fHxys3NNe9zbd9zoRj8/f115513av78+Ro7dqykvK1catWqZe6nfz41atTQ22+/bf5o4LJlyzR+/HiNGjVKNWrU0KOPPqo9e/ZI0nm3fElNTVXVqlWVmZmpcePGadasWTp06JDb/vSpqakXjMm1fVFRftwxNDS0wLivWrWqfvrpp/M+zjVmTp065Vbevn1788dEX3/99QJbDRX13JMK9r8rGXrVVVcVWu4aF8V5rc+nQYMGbrdtNpvq169v7r9eEs/jcDjUrl07rV27VlJeEv2mm27SjTfeqNzcXG3atEkhISE6fvy4W3J7z549+umnnxQUFFTocY8ePWrWS01NVXBw8HnruRTlnCstc+bM0Ztvvqldu3bpzJkzZnlh5+y57XS9xq52uq5d5/ahJMsP2C6Wj4+PsrOzC73v9OnT5gdHrv/PP+7z18tfBwAAlAyS6AAAAOfRpk2bAvvTurj2Gb755ps1depU1ahRQxUqVNCsWbMK/OhiZGSkQkJC9MEHH+jmm2/WBx98oOrVq7slhouqsCShJLek74X4+Pjo+++/16pVq7RkyRJ98803+uijj9S5c2d9++23bquRS0NJJHhcPxzoWil7Plbx5E8qv/baa/q///s/9e/fX2PHjlW1atVkt9s1bNiwQlcBW8XQp08fLVq0SBs2bNA111yjL774Qk888USxPpyw2Wxq2LChGjZsqKioKDVo0EDz5s3To48+arbl9ddfL7C/s4trVfeTTz6pWbNmadiwYWrXrp2qVKkim82mBx54oFgxFUVRXuPCNG7cWJK0Y8cOtx/gDQoKMs+PDz74wO0xxTn3zte2C7W5OK/1X1FSz3PjjTfq1Vdf1enTp7V27Vq9+OKL8vf319VXX621a9cqJCREktyS6E6nU7fccotGjhxZ6DEbNmxo1gsODnb7lkV+5ybhL3Y8/FUffPCB+vXrp+7du2vEiBEKDg6Ww+HQuHHjzA+FykM7C1OjRg3l5ubq6NGjbh9WZGdnKzk52fxdhmrVqsnLy0uHDx8ucAxX2bm/4QAAAP4akugAAAAX6eOPP5a3t7eWLVvm9rX6WbNmFajrcDj04IMPavbs2Ro/frw+++yzAltn1K5dW999912BH4TbtWuXeb90dqXkiRMn3J6jsJXqVgl3SbLb7erSpYu6dOmiCRMm6LXXXtOLL76oVatWFTu5X9S2l7Tc3NwCK5gv1uLFi9WpUyfNmDHDrfzEiRPmDzgWxW233aagoCDNmzdPbdu2VUZGhts3Doqrbt26qlq1qpkcq1evniTJz8/vgv20ePFi9e3b122l/unTpwuMHSuu59qxY4fq169/Ea2/sG7dusnhcGjevHl66KGHivSY4px7f0VxXuvzca00dzEMQ//73/907bXXFvt5zndO33TTTcrOztaCBQt06NAhM1l+8803m0n0hg0bmsl013OfOnXqgs9br149fffdd2rfvn25XuW8ePFi1a1bV5988onbazV69OiLOp7r2nVuH0rS7t27L66RFlwfoGzdulW33367Wb5161Y5nU7zfrvdrmuuuUZbt24tcIzNmzerbt26/KgoAAAljD3RAQAALpLD4ZDNZnNbAb5//3599tlnhdZ/5JFHlJKSooEDB+rUqVMF9si+/fbblZubq7feesutfOLEibLZbOrWrZukvERbYGBggRXYU6dOLfCcrv3Wz02aHj9+vEBdV4KmsC0CLqSobS9Jq1at0qlTp9xWL/8VDoejwOrTRYsW6dChQ8U6joeHh3r37q2FCxdq9uzZuuaaa8xk6fls3rxZ6enpBcq3bNmi5ORkNWrUSFLeCvx69erpjTfeKPQDhPx7xBcW03//+98if2vh1ltvVeXKlTVu3DhzmwiXklqpGxYWpv79++vrr78uMH6snqu4597FKs5rfT5z587VyZMnzduLFy/W4cOHzfOiOM9jdU5LeXu6V6hQQePHj1e1atXUrFkzSXnJ9U2bNmnNmjVuq9ClvH38N27cqGXLlhU43okTJ5STk2PWy83NNbcpyi8nJ6fIH8yUNtcHk/nHzObNm7Vx48aLOl6NGjXUokULzZkzx20LpOXLl+vXX3/9a409R+fOnVWtWjW9/fbbbuVvv/22KlasqKioKLOsZ8+e+uGHH9wS6bt379bKlSvVq1evEm0XAABgJToAAMBFi4qK0oQJE3TbbbfpwQcf1NGjRzVlyhTVr1+/0H2gr7vuOl199dVatGiRmjRpopYtW7rdf+edd6pTp0568cUXtX//fjVv3lzffvutPv/8cw0bNsxcrSpJjz76qP71r3/p0UcfVatWrfT999/rt99+K/CcERERkqQXX3xRDzzwgCpUqKA777xTY8aM0ffff6+oqCjVrl1bR48e1dSpUxUaGqobb7yx2K9Fcdp+MVJTU80tPXJycrR79269/fbb8vHx0XPPPfeXju1yxx13aMyYMYqOjtYNN9ygn3/+WfPmzVPdunWLfaw+ffroP//5j1atWqXx48cX6THvv/++5s2bp3vuuUcRERHy9PTUzp07NXPmTHl7e+uFF16QlLcKdfr06erWrZuaNWum6Oho1apVS4cOHdKqVavk5+enL7/80ozp/fffV5UqVdS0aVNt3LhR3333XaF7vBfGz89PEydO1KOPPqrWrVvrwQcfVNWqVbV9+3ZlZGRozpw5xX5tCjNp0iTFx8frySef1Icffqg777xTwcHBSkpK0vr16/Xll1+aHyJIxT/3LlZxXuvzqVatmm688UZFR0fryJEjmjRpkurXr6/HHnus2M9jdU77+vqqYsWKioiI0KZNm3TnnXeaK7FvvvlmpaenKz09vUASfcSIEfriiy90xx13qF+/foqIiFB6erp+/vlnLV68WPv371dgYKA6dOiggQMHaty4cYqLi9Ott96qChUqaM+ePVq0aJEmT56snj17lsjr/v3335sfEh47dkzp6en65z//acZy8803Wz72jjvu0CeffKJ77rlHUVFRio+P17Rp09S0adOL/tbKuHHjFBUVpRtvvFH9+/fX8ePH9d///lfNmjUr0jF///13vf/++5JkJr1d8dSuXdv8poqPj4/Gjh2rIUOGqFevXoqMjNTatWv1wQcf6NVXX1W1atXMYz7xxBN67733FBUVpX/84x+qUKGCJkyYoJCQED3zzDMXFScAADgPAwAAAAXMmjXLkGT88MMP5603Y8YMo0GDBoaXl5fRuHFjY9asWcbo0aMNq2nWv//9b0OS8dprrxV6/8mTJ42YmBijZs2aRoUKFYwGDRoYr7/+uuF0Ot3qZWRkGAMGDDCqVKliVK5c2bjvvvuMo0ePGpKM0aNHu9UdO3asUatWLcNutxuSjPj4eGPFihXG3XffbdSsWdPw9PQ0atasafTu3dv47bffLvjadOjQwWjWrNlFt12SMWTIkAs+T/7nk2T+s9lsRrVq1Yy77rrL2LZtm1tdV7/Fx8ebZbVr1zaioqIKPW6HDh3M26dPnzaeeeYZo0aNGoaPj4/Rvn17Y+PGjQXqrVq1ypBkLFq06LztbtasmWG3242DBw8WKc6ffvrJGDFihNGyZUujWrVqhoeHh1GjRg2jV69exo8//ligfmxsrHHvvfcaAQEBhpeXl1G7dm3jvvvuM1asWGHWSUlJMaKjo43AwECjUqVKRmRkpLFr1y6jdu3aRt++fc16FxrvX3zxhXHDDTcYPj4+hp+fn9GmTRtjwYIF5v1WY6Jv375G7dq1ixR/Tk6OMWvWLKNz585m/IGBgUaXLl2MadOmGZmZmW71i3ruFTbe4uPjDUnG66+/7lZu1bdFea0L4zreggULjOeff94IDg42fHx8jKioKOP3338vUL+oz1PYOe0yYsQIQ5Ixfvx4t8fUr1/fkGTs3bu3wPOePHnSeP7554369esbnp6eRmBgoHHDDTcYb7zxhpGdne1W99133zUiIiIMHx8fo3LlysY111xjjBw50khISDDrFPWcs+Lqx8L+nXt9O5fT6TRee+01o3bt2oaXl5dx3XXXGV999VWBsWg1BgzDKPR5Pv74Y6NJkya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      "text/plain": [
       "<Figure size 1500x600 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "\n",
    "plt.figure(figsize=(15, 6))\n",
    "positive_mask = np.array(payouts) >= 0\n",
    "negative_mask = np.array(payouts) < 0\n",
    "\n",
    "# Plot positive bars in blue and negative bars in red\n",
    "xs = np.r_[MIN_GUESS:MAX_GUESS+1]\n",
    "plt.bar(xs[positive_mask], np.array(payouts)[positive_mask], color='skyblue', edgecolor='navy')\n",
    "plt.bar(xs[negative_mask], np.array(payouts)[negative_mask], color='red', edgecolor='darkred')\n",
    "\n",
    "plt.axhline(y=average_payout, color='green', linestyle='--', label=f'Average: ${average_payout:.2f}')\n",
    "\n",
    "plt.title(f'Payouts for Binary Search Game between {MIN_GUESS} and {MAX_GUESS}')\n",
    "plt.xlabel('Target Number')\n",
    "plt.ylabel('Payout ($)')\n",
    "plt.legend()\n",
    "plt.grid(axis='y', linestyle='--', alpha=0.7)\n",
    "plt.gca().yaxis.set_major_formatter(plt.FuncFormatter(format_payout))\n",
    "\n",
    "plt.tight_layout()"
   ]
  }
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