2090:
20:
98:
467:
150:
771:
110:
units of fuel stored at a fixed base. The jeep can carry at most 1 unit of fuel at any time, and can travel 1 unit of distance on 1 unit of fuel (the jeep's fuel consumption is assumed to be constant). At any point in a trip the jeep may leave any amount of fuel that it is carrying at a fuel dump, or
462:
units on its final trip (the maximum distance traveled into the desert is half of this). It collects half of the remaining fuel at each dump on the way out, which fills its tank. After leaving the farthest fuel dump it travels 1/2 a unit further into the desert and then returns to the farthest fuel
101:
Plot to scale of the
Exploring (top) and Crossing (bottom) versions of the jeep problem for three units of fuel. The horizontal axis denotes distance and vertical axis denotes time. Vertical coloured line segments denote stashing fuel and horizontal ones denote travel using withdrawn fuel. Coloured
965:
495:. As the harmonic numbers are unbounded, it is possible to exceed any given distance on the final trip, as along as sufficient fuel is available at the base. However, the amount of fuel required and the number of fuel dumps both increase exponentially with the distance to be traveled.
457:
1122:
1280:
2162:; and go back to India. We had so little training on this problem of maximizing efficiency, we actually found to get some of the B-29s back instead of offloading fuel, they had to take it on. To make a long story short, it wasn't worth a damn. And it was
1799:
1486:
575:
59:
must maximize the distance it can travel into a desert with a given quantity of fuel. The jeep can only carry a fixed and limited amount of fuel, but it can leave fuel and collect fuel at fuel dumps anywhere in the desert.
128:
In either case the objective is to maximize the distance traveled by the jeep on its final trip. Alternatively, the objective may be to find the least amount of fuel required to produce a final trip of a given distance.
1363:
970:
so it is possible in theory to cross a desert of any size given enough fuel at the base. As before, the amount of fuel required and the number of fuel dumps both increase exponentially with the distance to be traveled.
2125:
that understanding the fuel efficiency issue caused by having to transport the fuel to forward bases was the main reason why the strategy of launching bombing raids from mainland China was abandoned in favor of the
785:
322:
2333:
Another common question is concerned with the maximum distance into a desert which could be reached from a frontier settlement by an explorer capable of carrying provisions that would last him for
776:
units on its final trip. It collects all of the remaining fuel at each dump on the way out, which fills its tank. After leaving the farthest fuel dump it travels a further distance of 1 unit.
992:
1129:
1857:
1999:
1680:
1538:
1282:, for crossing the desert where the jeep must return to the base at the end of every trip except for the final trip, when the jeep travels as far as it can before running out of fuel.
1955:
1906:
1636:
1587:
565: − 1 fuel dumps. The farthest contains 1/3 of a unit of fuel, the next farthest contain 1/5 of a unit of fuel, and so on, and the nearest fuel dump has just 1/(2
498:
The "crossing the desert" variant can be solved with a similar strategy, except that there is now no requirement to collect fuel on the way back on the final trip. So on trip
312: − 1 fuel dumps. The farthest contains 1/2 of a unit of fuel, the next farthest contain 1/3 of a unit of fuel, and so on, and the nearest fuel dump has just 1/
124: – the jeep must return to the base at the end of every trip except for the final trip, when the jeep travels as far as it can before running out of fuel.
1378:
The number of fuel units available at the base need not be an integer. In the general case, the maximum distance achievable for the "explore the desert" problem with
111:
may collect any amount of fuel that was left at a fuel dump on a previous trip, as long as its fuel load never exceeds 1 unit. There are two variants of the problem:
39:
versions of the jeep problem for three units of fuel – coloured arrows denote depots, diagonal segments denote travel and vertical segments denote fuel transfer
463:
dump. It collects the remaining fuel from each fuel dump on the way back, which is just enough to reach the next fuel dump or, in the final step, to return to base.
1703:
1390:
2099:, the attacking Vulcan was refuelled seven times on the outward journey and once on the return journey. Grey lines indicate reserve aircraft to replace casualties.
766:{\displaystyle 1+{\frac {1}{3}}+{\frac {1}{5}}+\cdots +{\frac {1}{2n-1}}=\sum _{k=1}^{n}{\frac {1}{2k-1}}=H_{2n-1}-{\frac {1}{2}}H_{n-1}\equiv \mathrm {cross} (n)}
2386:
569: − 1) units of fuel left. Together with 1 unit of fuel with which it starts from base, this means that the jeep can travel a total distance of
237: − 2) units of fuel from this second fuel dump on the way out, so that it leaves the fuel dump carrying 1 unit of fuel. It also collects 1/(2
474: = 3, showing fuel contents of jeep and fuel dumps at start of each trip, at turnaround point on first two trips, and at end of final trip.
316:
units of fuel left. Together with 1 unit of fuel with which it starts from base, this means that the jeep can travel a total round trip distance of
1290:
2134:"We had to fly those planes from the bases in Kansas to India. Then we had to fly fuel over the hump into China. We were supposed to take these
960:{\displaystyle \mathrm {cross} (n)=\sum _{k=1}^{n}{\frac {1}{2k-1}}>\sum _{k=1}^{n}{\frac {1}{2k}}={\frac {1}{2}}H_{n}=\mathrm {explore} (n)}
2015:
Note that the order of the jeep trips is not fixed. For example in the "exploring the desert" version of the problem, the jeep could make
241: − 2) units of fuel from the second fuel dump on the way back, which is just enough fuel to return to the first fuel dump.
141:
quantity. More complex variations on the problem have been proposed in which the fuel can only be left or collected in discrete amounts.
452:{\displaystyle 1+{\frac {1}{2}}+{\frac {1}{3}}+\cdots +{\frac {1}{n}}=\sum _{k=1}^{n}{\frac {1}{k}}\equiv 2\times \mathrm {explore} (n)}
199:) units of fuel from this first fuel dump on the way out, so that it leaves the fuel dump carrying 1 unit of fuel. It also collects 1/(2
2179:
1117:{\displaystyle \mathrm {explore} (n)={\frac {1}{2}}H_{n}={\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{6}}+\cdots +{\frac {1}{2n}}}
1275:{\displaystyle \mathrm {cross} (n)=H_{2n-1}-{\frac {1}{2}}H_{n-1}=1+{\frac {1}{3}}+{\frac {1}{5}}+\cdots +{\frac {1}{2n-1}}}
161:
A strategy that maximizes the distance traveled on the final trip for the "exploring the desert" variant is as follows:
2417:
157: = 3, showing fuel contents of jeep and fuel dumps at start of each trip and at turnround point on each trip.
2422:
2354:
2309:
65:
974:
In summary, the maximum distance reachable by the jeep (with a fuel capacity for 1 unit of distance at any time) in
222: − 2) units of fuel, which is just enough to return to the first fuel dump. Here it collects 1/(2
1807:
1960:
1641:
1494:
2234:
1911:
1862:
1592:
1543:
2094:
206:
On the second trip the jeep travels to the first fuel dump and refuels. It then travels a distance of 1/(2
2239:
203:) units of fuel from this first fuel dump on the way back, which is just enough fuel to return to base.
2195:
2167:
2211:
for better understanding on application of these ideas. In these missions, conducted during the
2150:; offload the fuel; fly back to India; make enough missions to build up fuel in Chengtu; fly to
2103:
The problem can have a practical application in wartime situations, especially with respect to
2301:
1794:{\displaystyle \mathrm {cross} (n)=\int _{0}^{n}{\frac {\mathrm {d} f}{2\lceil n-f\rceil -1}}}
1481:{\displaystyle \mathrm {explore} (n)=\int _{0}^{n}{\frac {\mathrm {d} f}{2\lceil n-f\rceil }}}
2293:
2208:
1124:, for exploring the desert where the jeep must return to the base at the end of every trip;
8:
138:
2398:
2350:
2305:
2294:
2265:
2338:
2228:
2224:
2066:
units of fuel are used to move fuel between the first and second fuel dump, using
218: − 1) units of fuel at a second fuel dump. The jeep still has 1/(2
2268:
2216:
2183:
2139:
2116:
2104:
1367:
483:
2342:
2289:
2127:
2121:
137:
In the classic problem the fuel in the jeep and at fuel dumps is treated as a
2411:
2212:
2187:
2089:
2039:-th trip one-way to the first fuel dump, thus arriving there with a total of
118: – the jeep must return to the base at the end of every trip.
1358:{\displaystyle H_{n}=1+{\frac {1}{2}}+{\frac {1}{3}}+\cdots +{\frac {1}{n}}}
2387:
Optimal
Logistics for Expeditions: the Jeep Problem with Complete Refilling
2163:
2108:
82:
2059:
units are saved for the return trip to base at the very end and the other
2220:
2159:
86:
1691:
The maximum distance achievable for the "cross the desert" problem with
2366:
19:
2273:
56:
77:, with the puzzle being about a travelling camel eating grain. The
2171:
1957:
units of distance away from the second fuel dump, and so on. Here
1908:
units of distance away from the first fuel dump, the third one at
1638:
units of distance away from the second fuel dump, and so on. Here
1589:
units of distance away from the first fuel dump, the third one at
2147:
1859:
units of distance away from the starting base, the second one at
1540:
units of distance away from the starting base, the second one at
466:
149:
2191:
2151:
514: + 1) units from the previous fuel dump and leaves (2
276: + 1) units of fuel there. On each of the subsequent
97:
74:
2022:
round-trips between the base and the first fuel dump, leaving
260: + 2) units from the previous fuel dump and leaves (
2155:
2143:
226:) units of fuel, which is just enough fuel to return to base.
85:
also discusses the problem. A modern treatment was given by
169:
trips. On each trip it starts from base with 1 unit of fuel.
2219:
used air to air refueling by staging tankers to enable the
2135:
2112:
2033:
units of fuel at the fuel dump each time and then make an
530: + 1) units of fuel there. On each of the next
63:
The problem first appeared in the 9th-century collection
2081:-th trip one-way to the second fuel dump. And so on.
188:) units of fuel – just enough to return to base.
2263:
1963:
1914:
1865:
1810:
1706:
1644:
1595:
1546:
1497:
1393:
1293:
1132:
995:
788:
578:
325:
184:
units of fuel at a fuel dump. The jeep still has 1/(2
172:
On the first trip the jeep travels a distance of 1/(2
102:
numbers denote units of fuel stashed at that moment.
561:Now when the jeep starts its final trip, there are
1993:
1949:
1900:
1851:
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1116:
959:
765:
451:
2182:at the end of World War II were flown using B-29
233: − 2 trips the jeep collects 1/(2
195: − 1 trips the jeep collects 1/(2
2409:
2166:who really came to that conclusion, and led the
2142:there. We were to fill them with fuel, fly from
244:The jeep continues in this way, so that on trip
558: + 1) units of fuel on its way back.
478:The distance travelled on the last trip is the
308:When the jeep starts its final trip, there are
153:Solution to "exploring the desert" variant for
470:Solution to "crossing the desert" variant for
304: + 2) units of fuel on its way back.
2107:. In the context of the bombing of Japan in
1988:
1982:
1970:
1964:
1935:
1929:
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1373:
2389:, Gunter Rote and Guochuan Zhang, June 1996
982:midway fuel dumps and consuming a total of
538: − 1 trips it collects 1/(2
16:Mathematical problem of placing fuel depots
1852:{\displaystyle \{n\}/(2\lceil n\rceil -1)}
2325:
2323:
2321:
2259:
2257:
2255:
2084:
1994:{\displaystyle \{n\}=n-\lfloor n\rfloor }
1675:{\displaystyle \{n\}=n-\lfloor n\rfloor }
2088:
1533:{\displaystyle \{n\}/(2\lceil n\rceil )}
465:
210: − 2) units and leaves (
148:
96:
18:
2377:, #475 (March 1992), pp. 102–126.
2288:
550:th dump on its way out and another 1/(2
546: + 1) units of fuel from the
296:th dump on its way out and another 1/(2
292: + 2) units of fuel from the
2410:
2318:
2296:My Best Mathematical and Logic Puzzles
2252:
1950:{\displaystyle 1/(2\lceil n\rceil -5)}
1901:{\displaystyle 1/(2\lceil n\rceil -3)}
1631:{\displaystyle 1/(2\lceil n\rceil -4)}
1582:{\displaystyle 1/(2\lceil n\rceil -2)}
2264:
2010:
2369:, John Hadley and David Singmaster,
1804:with the first fuel dump located at
1491:with the first fuel dump located at
55:is a mathematics problem in which a
2347:Mathematical Recreations and Essays
13:
2349:, Thirteenth Edition, Dover, p32.
1755:
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506:th fuel dump at a distance of 1/(2
436:
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252:th fuel dump at a distance of 1/(2
14:
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66:Propositiones ad Acuendos Juvenes
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2170:to move the whole thing to the
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2367:Problems to Sharpen the Young
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2235:Harmonic series (mathematics)
2051:units of fuel available. The
132:
71:Problems to Sharpen the Young
7:
2201:
502:the jeep establishes a new
144:
10:
2439:
2240:Optimization (mathematics)
2174:, which devastated Japan."
2001:is the fractional part of
1682:is the fractional part of
229:On each of the subsequent
191:On each of the subsequent
92:
2418:Mathematical optimization
2196:Northern Marianas Islands
1374:Continuous amount of fuel
2423:Recreational mathematics
2371:The Mathematical Gazette
2096:Operation Black Buck One
2073:round-trips and then an
522: − 1)/(2
27:vs distance from origin
2227:to bomb targets in the
2180:atomic bombing missions
214: − 2)/(
49:desert crossing problem
23:Plot of amount of fuel
2176:
2100:
2085:Practical applications
1995:
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284:trips it collects 1/(2
180: − 1)/
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103:
79:De viribus quantitatis
40:
2401:, www.errolmorris.com
2399:Fog of War transcript
2331:Exploration problems.
2132:
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248:it establishes a new
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2209:Operation Black Buck
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554: − 2
542: − 2
526: − 2
518: − 2
510: − 2
323:
300: − 2
288: − 2
256: − 2
176:) units and leaves (
116:Exploring the desert
1750:
1443:
534: −
280: −
272: −
264: −
122:Crossing the desert
53:exploration problem
2300:. Dover. pp.
2266:Weisstein, Eric W.
2101:
2011:Order independence
1991:
1947:
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1849:
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986:units of fuel) is
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159:
104:
41:
2223:bombers based on
2119:says in the film
1789:
1697:units of fuel is
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1384:units of fuel is
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2339:W. W. Rouse Ball
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2225:Ascension Island
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2290:Gardner, Martin
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2217:Royal Air Force
2204:
2184:Superfortresses
2140:tanker aircraft
2138:—there were no
2117:Robert McNamara
2105:fuel efficiency
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33:exploring (1–3)
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2269:"Jeep Problem"
2250:
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2128:island hopping
2122:The Fog of War
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1557:
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1549:
1529:
1526:
1523:
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1517:
1514:
1510:
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1500:
1489:
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1471:
1468:
1465:
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1441:
1436:
1432:
1428:
1425:
1422:
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1397:
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1197:
1191:
1188:
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1125:
1110:
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1103:
1098:
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1079:
1074:
1071:
1066:
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1058:
1053:
1048:
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1038:
1035:
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1027:
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1021:
1017:
1014:
1011:
1008:
1005:
1002:
999:
968:
967:
956:
953:
950:
946:
943:
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934:
931:
928:
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919:
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895:
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888:
881:
876:
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856:
853:
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847:
843:
836:
831:
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821:
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814:
811:
808:
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798:
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773:
762:
759:
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749:
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736:
731:
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721:
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712:
707:
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696:
693:
689:
685:
679:
676:
673:
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666:
659:
654:
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640:
634:
631:
628:
625:
621:
616:
613:
610:
605:
602:
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592:
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584:
581:
490:
460:
459:
448:
445:
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420:
416:
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405:
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395:
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371:
368:
363:
360:
357:
352:
349:
344:
339:
336:
331:
328:
306:
305:
242:
227:
204:
189:
170:
146:
143:
134:
131:
126:
125:
119:
94:
91:
15:
9:
6:
4:
3:
2:
2435:
2424:
2421:
2419:
2416:
2415:
2413:
2400:
2395:
2388:
2383:
2376:
2372:
2368:
2363:
2356:
2355:0-486-25357-0
2352:
2348:
2344:
2340:
2336:
2332:
2326:
2324:
2322:
2313:
2311:0-486-28152-3
2307:
2303:
2298:
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2285:
2276:
2275:
2270:
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2260:
2258:
2256:
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2238:
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2233:
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2222:
2218:
2214:
2213:Falklands War
2210:
2206:
2205:
2199:
2197:
2193:
2189:
2185:
2181:
2175:
2173:
2169:
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2153:
2149:
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2131:
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2124:
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2110:
2106:
2098:
2097:
2091:
2082:
2078:
2070:
2063:
2056:
2048:
2044:
2037:
2031:
2027:
2019:
2008:
2005:
1985:
1979:
1976:
1973:
1967:
1941:
1938:
1932:
1926:
1919:
1915:
1892:
1889:
1883:
1877:
1870:
1866:
1843:
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1814:
1785:
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1746:
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1733:
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1700:
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1648:
1622:
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1596:
1573:
1570:
1564:
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1551:
1547:
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1469:
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1463:
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1387:
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1234:
1229:
1224:
1221:
1216:
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1210:
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1199:
1195:
1189:
1186:
1181:
1176:
1173:
1170:
1167:
1163:
1159:
1153:
1126:
1108:
1105:
1101:
1096:
1093:
1090:
1085:
1082:
1077:
1072:
1069:
1064:
1059:
1056:
1051:
1046:
1042:
1036:
1033:
1028:
1022:
989:
988:
987:
985:
981:
977:
972:
951:
922:
917:
913:
907:
904:
899:
893:
890:
886:
879:
874:
871:
868:
864:
860:
854:
851:
848:
845:
841:
834:
829:
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823:
819:
815:
809:
782:
781:
780:
777:
757:
734:
729:
726:
723:
719:
713:
710:
705:
700:
697:
694:
691:
687:
683:
677:
674:
671:
668:
664:
657:
652:
649:
646:
642:
638:
632:
629:
626:
623:
619:
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611:
608:
603:
600:
595:
590:
587:
582:
579:
572:
571:
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564:
559:
557:
553:
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545:
541:
537:
533:
529:
525:
521:
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493:
489:
485:
481:
473:
468:
464:
443:
414:
411:
408:
403:
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393:
388:
385:
382:
378:
374:
369:
366:
361:
358:
355:
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347:
342:
337:
334:
329:
326:
319:
318:
317:
315:
311:
303:
299:
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291:
287:
283:
279:
275:
271:
267:
263:
259:
255:
251:
247:
243:
240:
236:
232:
228:
225:
221:
217:
213:
209:
205:
202:
198:
194:
190:
187:
183:
179:
175:
171:
168:
164:
163:
162:
156:
151:
142:
140:
130:
123:
120:
117:
114:
113:
112:
109:
99:
90:
88:
84:
81:(c. 1500) of
80:
76:
72:
68:
67:
61:
58:
54:
50:
46:
38:
34:
30:
26:
21:
2394:
2382:
2374:
2370:
2362:
2346:
2334:
2330:
2295:
2284:
2272:
2177:
2133:
2120:
2109:World War II
2102:
2095:
2076:
2068:
2061:
2054:
2046:
2042:
2035:
2029:
2025:
2017:
2014:
2003:
1803:
1693:
1690:
1684:
1490:
1380:
1377:
1287:
1285:
983:
979:
978:trips (with
975:
973:
969:
778:
775:
566:
562:
560:
555:
551:
547:
543:
539:
535:
531:
527:
523:
519:
515:
511:
507:
503:
499:
497:
491:
487:
479:
477:
471:
461:
313:
309:
307:
301:
297:
293:
289:
285:
281:
277:
273:
269:
265:
261:
257:
253:
249:
245:
238:
234:
230:
223:
219:
215:
211:
207:
200:
196:
192:
185:
181:
177:
173:
166:
160:
154:
136:
127:
121:
115:
107:
105:
83:Luca Pacioli
78:
70:
64:
62:
52:
48:
45:jeep problem
44:
42:
36:
32:
28:
24:
2160:steel mills
2158:; bomb the
2045:− 1) + 1/(2
1366:is the nth
2412:Categories
2246:References
2190:island of
2130:strategy:
779:Note that
139:continuous
133:Variations
106:There are
87:N. J. Fine
2274:MathWorld
2186:from the
1989:⌋
1983:⌊
1980:−
1939:−
1936:⌉
1930:⌈
1890:−
1887:⌉
1881:⌈
1841:−
1838:⌉
1832:⌈
1783:−
1780:⌉
1774:−
1768:⌈
1738:∫
1670:⌋
1664:⌊
1661:−
1620:−
1617:⌉
1611:⌈
1571:−
1568:⌉
1562:⌈
1525:⌉
1519:⌈
1473:⌉
1467:−
1461:⌈
1431:∫
1340:⋯
1264:−
1246:⋯
1203:−
1182:−
1174:−
1094:⋯
865:∑
852:−
820:∑
735:≡
727:−
706:−
698:−
675:−
643:∑
630:−
612:⋯
415:×
409:≡
379:∑
359:⋯
89:in 1947.
2345:(1987).
2292:(1994).
2202:See also
2172:Marianas
145:Solution
2337:days."
2194:in the
2188:Pacific
2148:Chengtu
2028:− 1) /
93:Problem
2353:
2308:
2221:Vulcan
2215:, the
2192:Tinian
2168:Chiefs
2152:Yawata
75:Alcuin
2207:Read
2178:(The
2164:LeMay
2156:Japan
2144:India
2136:B-29s
2113:B-29s
1286:Here
2351:ISBN
2341:and
2306:ISBN
2053:1/(2
861:>
57:jeep
43:The
35:and
31:for
2198:.)
2146:to
2111:by
2093:In
2079:−1)
2071:− 2
2064:− 1
2020:− 1
980:n-1
482:th
268:)/(
51:or
2414::
2375:76
2373:,
2320:^
2304:.
2302:53
2271:.
2254:^
2154:,
2115:,
2007:.
1688:.
1370:.
486:,
47:,
2357:.
2335:a
2329:"
2314:.
2277:.
2231:.
2077:n
2075:(
2069:n
2062:n
2057:)
2055:n
2049:)
2047:n
2043:n
2041:(
2036:n
2030:n
2026:n
2024:(
2018:n
2004:n
1986:n
1977:n
1974:=
1971:}
1968:n
1965:{
1945:)
1942:5
1933:n
1927:2
1924:(
1920:/
1916:1
1896:)
1893:3
1884:n
1878:2
1875:(
1871:/
1867:1
1847:)
1844:1
1835:n
1829:2
1826:(
1822:/
1818:}
1815:n
1812:{
1786:1
1777:f
1771:n
1765:2
1760:f
1756:d
1747:n
1742:0
1734:=
1731:)
1728:n
1725:(
1721:s
1718:s
1715:o
1712:r
1709:c
1694:n
1685:n
1667:n
1658:n
1655:=
1652:}
1649:n
1646:{
1626:)
1623:4
1614:n
1608:2
1605:(
1601:/
1597:1
1577:)
1574:2
1565:n
1559:2
1556:(
1552:/
1548:1
1528:)
1522:n
1516:2
1513:(
1509:/
1505:}
1502:n
1499:{
1470:f
1464:n
1458:2
1453:f
1449:d
1440:n
1435:0
1427:=
1424:)
1421:n
1418:(
1414:e
1411:r
1408:o
1405:l
1402:p
1399:x
1396:e
1381:n
1351:n
1348:1
1343:+
1337:+
1332:3
1329:1
1324:+
1319:2
1316:1
1311:+
1308:1
1305:=
1300:n
1296:H
1267:1
1261:n
1258:2
1254:1
1249:+
1243:+
1238:5
1235:1
1230:+
1225:3
1222:1
1217:+
1214:1
1211:=
1206:1
1200:n
1196:H
1190:2
1187:1
1177:1
1171:n
1168:2
1164:H
1160:=
1157:)
1154:n
1151:(
1147:s
1144:s
1141:o
1138:r
1135:c
1109:n
1106:2
1102:1
1097:+
1091:+
1086:6
1083:1
1078:+
1073:4
1070:1
1065:+
1060:2
1057:1
1052:=
1047:n
1043:H
1037:2
1034:1
1029:=
1026:)
1023:n
1020:(
1016:e
1013:r
1010:o
1007:l
1004:p
1001:x
998:e
984:n
976:n
955:)
952:n
949:(
945:e
942:r
939:o
936:l
933:p
930:x
927:e
923:=
918:n
914:H
908:2
905:1
900:=
894:k
891:2
887:1
880:n
875:1
872:=
869:k
855:1
849:k
846:2
842:1
835:n
830:1
827:=
824:k
816:=
813:)
810:n
807:(
803:s
800:s
797:o
794:r
791:c
761:)
758:n
755:(
751:s
748:s
745:o
742:r
739:c
730:1
724:n
720:H
714:2
711:1
701:1
695:n
692:2
688:H
684:=
678:1
672:k
669:2
665:1
658:n
653:1
650:=
647:k
639:=
633:1
627:n
624:2
620:1
615:+
609:+
604:5
601:1
596:+
591:3
588:1
583:+
580:1
567:n
563:n
556:k
552:n
548:k
544:k
540:n
536:k
532:n
528:k
524:n
520:k
516:n
512:k
508:n
504:k
500:k
492:n
488:H
480:n
472:n
447:)
444:n
441:(
437:e
434:r
431:o
428:l
425:p
422:x
419:e
412:2
404:k
401:1
394:n
389:1
386:=
383:k
375:=
370:n
367:1
362:+
356:+
351:3
348:1
343:+
338:2
335:1
330:+
327:1
314:n
310:n
302:k
298:n
294:k
290:k
286:n
282:k
278:n
274:k
270:n
266:k
262:n
258:k
254:n
250:k
246:k
239:n
235:n
231:n
224:n
220:n
216:n
212:n
208:n
201:n
197:n
193:n
186:n
182:n
178:n
174:n
167:n
155:n
108:n
69:(
29:d
25:f
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