How Many Food Packets Did the Friends Bring on Their Camping Trip?

[Icebreaker]

[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.

 

[BicycleTree]

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.
[/color]

 

[BicycleTree]

Although no camper is likely to bring THAT much food, unless the packets are very very small.

 

[Icebreaker]

I forgot the word "minimum".

Btw, I randomly tested a few of your answers, and they were all wrong.

 

[BicycleTree]

They all seem to be of the form 3121 + 3125n, though I couldn't tell you why.

 

[BicycleTree]

I forgot the word "minimum".

Btw, I randomly tested a few of your answers, and they were all wrong.

Seriously? Hold on let me check.

 

[Icebreaker]

Although no camper is likely to bring THAT much food, unless the packets are very very small.

The number does not matter as long as it fits the conditions. Think of it as a math problem in disguise.

 

[BicycleTree]

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

 

[medshredr]

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.

 

[BicycleTree]

By the way, I solved this by brute force (wrote a program). I'm interested to know how you would do it algebraically.

 

[Icebreaker]

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.

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.

 

[BicycleTree]

This is my program:

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 + ", ");
		}
	}
}

 

[Icebreaker]

$$m = \frac{4}{5}n-\sum_{i = 0}^{5}\frac{4^i}{5^i}$$, where $$\{m, n\}$$ are integers.

The graph will show all possible solutions.

The other "elegant" solution is -4.

 

[ArielGenesis]

bicycletree, you seems to be familiar with c ++

 

[quark]

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.

 

[BicycleTree]

bicycletree, you seems to be familiar with c ++

Well, I have had a course in C++ but right now I use Java.

 

[BicycleTree]

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.

They all are of the form 3121 (mod 3125). Now where does the 3121 come from?

 

[quark]

Sorry, I meant if (256x-2101)625 is perfectly divisible by 5(quotient should be zero). Trial and error method will yield the result.

 

[Icebreaker]

Was my algebraic expression correct?

I prefer the answer of: -4.

 

[BicycleTree]

Sorry, I meant if (256x-2101)625 is perfectly divisible by 5(quotient should be zero). Trial and error method will yield the result.

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.

 

[Rahmuss]

bicycletree, you seems to be familiar with c ++

That's actually java, not C++. C++ would use "count<<" instead of "System.out.print".

I still don't see how it could ever be a possibility for the packets to be divided if you can't sub-divide them. It doesn't matter how large a number you have, if it's divisible by five, then it ends in a 5 or a 0, and then if you subract one from that number, then you'll have a number that is NOT divisible by 5. Please explain.

 

[quark]

Rahmuss,

You might have not read the question clearly. First take one out of the initial qty and then divide it by 5(fair chances for a perfect division). Take one part out of it and then take one out of it and then again divide by 5.

For ex. 21-1 = 20(divisible by 5), take one part, so 16 remains, take one out of it, 15 remains(again divisible by 5) and so on.

BicycleTree,

Last and final correction to my post is, if (256x-2101)/625 is perfectly divisible by 5 and you are right. Probably, we can get your equation only after we know the answers. We will just see if anybody comes up with a solution, without having to use a program.

Hmmm...right time to read Trachtenberg's Speed System of Calculations.