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The Bear and the Food

  1. Apr 23, 2005 #1
    [SOLVED] The Bear and the Food

    5 friends go camping, and bring with them a number of packets of food. In the middle of the night, one of them wakes up, sees a hungry bear, and decides to give the bear one packet of food. He then proceeds to divide the remaining packets of food into 5 equal parts, and takes one part for himself, then goes back to bed. A second friend wakes up, sees the bear, gives the bear a packet of food, then proceeds to, as with the first, divide the remaining food into 5 parts and keeps one for himself. The third, fourth and fifth each wakes up one after another and does the same thing. What is the minimum packets of food in the beginning.

    There are 2 possible answers that I'm aware of.
     
    Last edited by a moderator: Apr 23, 2005
  2. jcsd
  3. Apr 23, 2005 #2
    There are a number of possible answers that I'm aware of:

    3121, 6246, 9371, 12496, 15621, 18746, 21871, 24996, 28121, 31246, 34371, 37496, 40621, 43746, 46871, 49996, 53121, 56246, 59371, 62496, 65621, 68746, 71871, 74996, 78121, 81246, 84371, 87496, 90621, 93746, 96871, 99996, 103121, 106246, 109371, 112496, 115621, 118746, 121871, 124996, 128121, 131246, 134371, 137496, 140621, 143746, 146871, 149996, 153121, 156246, 159371, 162496, 165621, 168746, 171871, 174996, 178121, 181246, 184371, 187496, 190621, 193746, 196871, 199996, 203121, 206246, 209371, 212496, 215621, 218746, 221871, 224996, 228121, 231246, 234371, 237496, 240621, 243746, 246871, 249996, 253121, 256246, 259371, 262496, 265621, 268746, 271871, 274996, 278121, 281246, 284371, 287496, 290621, 293746, 296871, 299996, 303121, 306246, 309371, 312496, 315621, 318746, 321871, 324996, 328121, 331246, 334371, 337496, 340621, 343746, 346871, 349996, 353121, 356246, 359371, 362496, 365621, 368746, 371871, 374996, 378121, 381246, 384371, 387496, 390621, 393746, 396871, 399996, 403121, 406246, 409371, 412496, 415621, 418746, 421871, 424996, 428121, 431246, 434371, 437496, 440621, 443746, 446871, 449996, 453121, 456246, 459371, 462496, 465621, 468746, 471871, 474996, 478121, 481246, 484371, 487496, 490621, 493746, 496871, 499996, 503121, 506246, 509371, 512496, 515621, 518746, 521871, 524996, 528121, 531246, 534371, 537496, 540621, 543746, 546871, 549996, 553121, 556246, 559371, 562496, 565621, 568746, 571871, 574996, 578121, 581246, 584371, 587496, 590621, 593746, 596871, 599996, 603121, 606246, 609371, 612496, 615621, 618746, 621871, 624996, 628121, 631246, 634371, 637496, 640621, 643746, 646871, 649996, 653121, 656246, 659371, 662496, 665621, 668746, 671871, 674996, 678121, 681246, 684371, 687496, 690621, 693746, 696871, 699996, 703121, 706246, 709371, 712496, 715621, 718746, 721871, 724996, 728121, 731246, 734371, 737496, 740621, 743746, 746871, 749996, 753121, 756246, 759371, 762496, 765621, 768746, 771871, 774996, 778121, 781246, 784371, 787496, 790621, 793746, 796871, 799996, 803121, 806246, 809371, 812496, 815621, 818746, 821871, 824996, 828121, 831246, 834371, 837496, 840621, 843746, 846871, 849996, 853121, 856246, 859371, 862496, 865621, 868746, 871871, 874996, 878121, 881246, 884371, 887496, 890621, 893746, 896871, 899996, 903121, 906246, 909371, 912496, 915621, 918746, 921871, 924996, 928121, 931246, 934371, 937496, 940621, 943746, 946871, 949996, 953121, 956246, 959371, 962496, 965621, 968746, 971871, 974996, 978121, 981246, 984371, 987496, 990621, 993746, 996871, 999996, 1003121, 1006246, 1009371, 1012496, 1015621, 1018746, 1021871, 1024996, 1028121, 1031246, 1034371, 1037496, 1040621, 1043746, 1046871, 1049996, 1053121, 1056246, 1059371, 1062496, 1065621, 1068746, 1071871, 1074996, 1078121, 1081246, 1084371, 1087496, 1090621, 1093746, 1096871, 1099996, 1103121, 1106246, 1109371, 1112496, 1115621, 1118746, 1121871, 1124996, 1128121, 1131246, 1134371, 1137496, 1140621, 1143746, 1146871, 1149996, 1153121, 1156246, 1159371, 1162496, 1165621, 1168746, 1171871, 1174996, 1178121, 1181246, 1184371, 1187496, 1190621, 1193746, 1196871, 1199996, 1203121, 1206246, 1209371, 1212496, 1215621, 1218746, 1221871, 1224996, 1228121, 1231246, 1234371, 1237496, 1240621, 1243746, 1246871, 1249996, 1253121, 1256246, 1259371, 1262496, 1265621, 1268746, 1271871, 1274996, 1278121, 1281246, 1284371, 1287496, 1290621, 1293746, 1296871, 1299996, 1303121, 1306246, 1309371, 1312496, 1315621, 1318746, 1321871, 1324996, 1328121, 1331246, 1334371, 1337496, 1340621, 1343746, 1346871, 1349996, 1353121, 1356246, 1359371, 1362496, 1365621, 1368746, 1371871, 1374996, 1378121, 1381246, 1384371, 1387496, 1390621, 1393746, 1396871, 1399996, 1403121, 1406246, 1409371, 1412496, 1415621, 1418746, 1421871, 1424996, 1428121, 1431246, 1434371, 1437496, 1440621, 1443746, 1446871, 1449996, 1453121, 1456246, 1459371, 1462496, 1465621, 1468746, 1471871, 1474996, 1478121, 1481246, 1484371, 1487496, 1490621, 1493746, 1496871, 1499996, 1503121, 1506246, 1509371, 1512496, 1515621, 1518746, 1521871, 1524996, 1528121, 1531246, 1534371, 1537496, 1540621, 1543746, 1546871, 1549996, 1553121, 1556246, 1559371, 1562496, 1565621, 1568746, 1571871, 1574996, 1578121, 1581246, 1584371, 1587496, 1590621, 1593746, 1596871, 1599996, 1603121, 1606246, 1609371, 1612496, 1615621, 1618746, 1621871, 1624996, 1628121, 1631246, 1634371, 1637496, 1640621, 1643746, 1646871, 1649996, 1653121, 1656246, 1659371, 1662496, 1665621, 1668746, 1671871, 1674996, 1678121, 1681246, 1684371, 1687496, 1690621, 1693746, 1696871, 1699996, 1703121, 1706246, 1709371, 1712496, 1715621, 1718746, 1721871, 1724996, 1728121, 1731246, 1734371, 1737496, 1740621, 1743746, 1746871, 1749996, 1753121, 1756246, 1759371, 1762496, 1765621, 1768746, 1771871, 1774996, 1778121, 1781246, 1784371, 1787496, 1790621, 1793746, 1796871, 1799996, 1803121, 1806246, 1809371, 1812496, 1815621, 1818746, 1821871, 1824996, 1828121, 1831246, 1834371, 1837496, 1840621, 1843746, 1846871, 1849996, 1853121, 1856246, 1859371, 1862496, 1865621, 1868746, 1871871, 1874996, 1878121, 1881246, 1884371, 1887496, 1890621, 1893746, 1896871, 1899996, 1903121, 1906246, 1909371, 1912496, 1915621, 1918746, 1921871, 1924996, 1928121, 1931246, 1934371, 1937496, 1940621, 1943746, 1946871, 1949996, 1953121, 1956246, 1959371, 1962496, 1965621, 1968746, 1971871, 1974996, 1978121, 1981246, 1984371, 1987496, 1990621, 1993746, 1996871, 1999996, 2003121, 2006246, 2009371, 2012496, 2015621, 2018746, 2021871, 2024996, 2028121, 2031246, 2034371, 2037496, 2040621, 2043746, 2046871, 2049996, 2053121, 2056246, 2059371, 2062496, 2065621, 2068746, 2071871, 2074996, 2078121, 2081246, 2084371, 2087496, 2090621, 2093746, 2096871, 2099996, 2103121, 2106246, 2109371, 2112496, 2115621, 2118746, 2121871, 2124996, 2128121, 2131246, 2134371, 2137496, 2140621, 2143746, 2146871, 2149996, 2153121, 2156246, 2159371, 2162496, 2165621, 2168746, 2171871, 2174996, 2178121, 2181246, 2184371, 2187496, 2190621, 2193746, 2196871, 2199996, 2203121, 2206246, 2209371, 2212496, 2215621, 2218746, 2221871, 2224996, 2228121, 2231246, 2234371, 2237496, 2240621, 2243746, 2246871, 2249996, 2253121, 2256246, 2259371, 2262496, 2265621, 2268746, 2271871, 2274996, 2278121, 2281246, 2284371, 2287496, 2290621, 2293746, 2296871, 2299996, 2303121, 2306246, 2309371, 2312496, 2315621, 2318746, 2321871, 2324996, 2328121, 2331246, 2334371, 2337496, 2340621, 2343746, 2346871, 2349996, 2353121, 2356246, 2359371, 2362496, 2365621, 2368746, 2371871, 2374996, 2378121, 2381246, 2384371, 2387496, 2390621, 2393746, 2396871, 2399996, 2403121, 2406246, 2409371, 2412496, 2415621, 2418746, 2421871, 2424996, 2428121, 2431246, 2434371, 2437496, 2440621, 2443746, 2446871, 2449996, 2453121, 2456246, 2459371, 2462496, 2465621, 2468746, 2471871, 2474996, 2478121, 2481246, 2484371, 2487496, 2490621, 2493746, 2496871, 2499996, 2503121, 2506246, 2509371, 2512496, 2515621, 2518746, 2521871, 2524996, 2528121, 2531246, 2534371, 2537496, 2540621, 2543746, 2546871, 2549996, 2553121, 2556246, 2559371, 2562496, 2565621, 2568746, 2571871, 2574996, 2578121, 2581246, 2584371, 2587496, 2590621, 2593746, 2596871, 2599996, 2603121, 2606246, 2609371, 2612496, 2615621, 2618746, 2621871, 2624996, 2628121, 2631246, 2634371, 2637496, 2640621, 2643746, 2646871, 2649996, 2653121, 2656246, 2659371, 2662496, 2665621, 2668746, 2671871, 2674996, 2678121, 2681246, 2684371, 2687496, 2690621, 2693746, 2696871, 2699996, 2703121, 2706246, 2709371, 2712496, 2715621, 2718746, 2721871, 2724996, 2728121, 2731246, 2734371, 2737496, 2740621, 2743746, 2746871, 2749996, 2753121, 2756246, 2759371, 2762496, 2765621, 2768746, 2771871, 2774996, 2778121, 2781246, 2784371, 2787496, 2790621, 2793746, 2796871, 2799996, 2803121, 2806246, 2809371, 2812496, 2815621, 2818746, 2821871, 2824996, 2828121, 2831246, 2834371, 2837496, 2840621, 2843746, 2846871, 2849996, 2853121, 2856246, 2859371, 2862496, 2865621, 2868746, 2871871, 2874996, 2878121, 2881246, 2884371, 2887496, 2890621, 2893746, 2896871, 2899996, 2903121, 2906246, 2909371, 2912496, 2915621, 2918746, 2921871, 2924996, 2928121, 2931246, 2934371, 2937496, 2940621, 2943746, 2946871, 2949996, 2953121, 2956246, 2959371, 2962496, 2965621, 2968746, 2971871, 2974996, 2978121, 2981246, 2984371, 2987496, 2990621, 2993746, 2996871, 2999996, 3003121, 3006246, 3009371, 3012496, 3015621, 3018746, 3021871, 3024996, 3028121, 3031246, 3034371, 3037496, 3040621, 3043746, 3046871, 3049996.
     
  4. Apr 23, 2005 #3
    Although no camper is likely to bring THAT much food, unless the packets are very very small.
     
  5. Apr 23, 2005 #4
    I forgot the word "minimum".

    Btw, I randomly tested a few of your answers, and they were all wrong.
     
  6. Apr 23, 2005 #5
    They all seem to be of the form 3121 + 3125n, though I couldn't tell you why.
     
  7. Apr 23, 2005 #6
    Seriously? Hold on let me check.
     
  8. Apr 23, 2005 #7
    The number does not matter as long as it fits the conditions. Think of it as a math problem in disguise.
     
  9. Apr 23, 2005 #8
    Well, I'm thinking that the "packets" are peanut or cashew shells.

    I checked 3121 and it does work.
    Beginning:
    3121 cashews
    After camper 1:
    (3121 - 1)*4/5 = 2496
    After camper 2:
    (2496 - 1)*4/5 = 1996
    After camper 3:
    (1996 - 1)*4/5 = 1596
    After camper 4:
    (1596 - 1)*4/5 = 1276
    After camper 5:
    (1276 - 1)*4/5 = 1020
     
  10. Apr 23, 2005 #9
    If the contents of each individual packet cannot be divided and resealed, at some point the remaining packets will not be divisible by 5 and this teaser does not work. Assuming the individual contents can be divided and resealed, any number other than 1 will work.
     
  11. Apr 23, 2005 #10
    By the way, I solved this by brute force (wrote a program). I'm interested to know how you would do it algebraically.
     
  12. Apr 23, 2005 #11
    Well I used the word "packets" hoping that it would invoke the word "quanta". Anyway, the packets cannot be subdivided.

    I tested the first few answers, and they seem to be right.

    I'll post an algebraic solution later. But even with the word "minimum", there are still 2 possible answers. One is more "abstract" than the other.
     
  13. Apr 23, 2005 #12
    This is my program:
    Code (Text):

    class BearFood{
        public static void main(String arg[])
        {
            int n;
            boolean stillok=true;
            for(int i = 1; i < 1000000; i++)
            {
                n = i;
                stillok=true;
                for(int j = 1; j <=5; j++)
                {
                    if(n % 4 == 0)
                        n = n*5/4 + 1;
                    else{
                        stillok=false;
                        break;
                    }
                }
                if(stillok==true)
                    System.out.print(n + ", ");
            }
        }
    }
     
     
  14. Apr 23, 2005 #13
    [tex]m = \frac{4}{5}n-\sum_{i = 0}^{5}\frac{4^i}{5^i}[/tex], where [tex]\{m, n\}[/tex] are integers.

    The graph will show all possible solutions.

    The other "elegant" solution is -4.
     
    Last edited by a moderator: Apr 23, 2005
  15. Apr 26, 2005 #14
    bicycletree, you seems to be familiar with c ++
     
  16. Apr 26, 2005 #15
    Did the lengthy algebraic expression. If quotient of (256x-2101)/625 when divided by 5 is a natural number, you will get the answer. Above given numbers fit well.
     
  17. Apr 26, 2005 #16
    Well, I have had a course in C++ but right now I use Java.
     
  18. Apr 26, 2005 #17
    They all are of the form 3121 (mod 3125). Now where does the 3121 come from?
     
  19. Apr 26, 2005 #18
    Sorry, I meant if (256x-2101)625 is perfectly divisible by 5(quotient should be zero). Trial and error method will yield the result.
     
  20. Apr 26, 2005 #19
    Was my algebraic expression correct?

    I prefer the answer of: -4.
     
  21. Apr 26, 2005 #20
    No, you meant what you originally said. Any integer would fit (256x-2101)625.

    I wasn't referring to your equation when I said they are all of the form 3121 (mod 3125). That was something I noticed in the list my program made.

    Icebreaker, I don't think your equation works. The sum from i = 0 to 5 of 4^i/5^i
    1 + 4/5 + 16/25 + 64/125 + 256/625 + 1024/3125 = 3.68928. If n is an integer then 4/5n will be all zeroes once you go two places or more after the decimal point. So if you subtract then 3.68928 you will always get a non-integer.
     
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