The North Pole Expedition

James Henry

Arizona Teacher Institute

Mathematics Project Report

In partial fulfillment of the M.A. Degree in Middle School MathematicsTeaching Leadership

Department of Mathematics

University of Arizona

April 2010

March to the North Pole1:

There is a group at their final base camp 50 miles from the North Pole. Each individual at the

camp can only carry enough provisions to continue on for 20 miles. This would allow one to travel 10

miles out before having to return to base-camp. Being that it is so cold, the travelers can store

provisions in the ice to be used by others. One person could not reach the North Pole on his/her own.

But, being that provisions can be stored for future use, if they work together as a team, they can

succeed in getting to the North Pole and back. The key would be to store provisions in the ice and add

more legs to the trip to extend the distance they can travel before having to return home.

How Adding Legs and Travelers Impacts the Expedition:

This illustration is a representation of an expedition with people that will be traveling in increasing

increments of 5 miles. In this case each additional leg adds 5 miles to the previous leg. If there is one

traveler to set out, as in the 1st leg, that person can go out 10 miles, using half his/her supplies, and then

have to return back to base camp. Adding more travelers and more legs allows the expedition to extend

1 Stevenson, F.W. Exploratory Problems in Mathematics. National Council of Teachers of Mathematics, 1992 pp. 117-118

1

further. Throughout the paper, in the illustrations, the up arrows represent a person traveling away from

the base camp and towards the North Pole, while the down arrows represent the return trip of that same

person. A leg will refer to the breaking down of the expedition into smaller increments and an addition

of distance traveled. The number of travelers needed to accomplish the legs will vary greatly. In this

scenario of legs with a constant increase of 5 miles, the use of 3 legs requires four travelers to

accomplish:

•The first and second persons travel 5 miles, leave 10 provisions2 (with two travelers this means20 provisions left at this point), and return 5 miles.

•The third person travels 5 miles, picks up 5 of the 20 dropped provisions, continues 5 more miles,drops 10 provisions at another drop point, returns 5 miles, and collects 5 more of the 20 droppedprovisions to get home. There are now 10 provisions left at the first drop point and 10 provisions atthe second drop point.

•The fourth person travels 5 miles, picks up 5 provisions at the first drop point, travels 5 miles,picks up 5 provisions at the second drop point (now has 20 miles of supplies), travels out and back10 miles, picks up the other 5 provisions at the second drop point to get to the first drop point,picks up the remaining 5 provisions at the original drop point to get back to camp.

The use of a 4th leg requires eight travelers to take the same idea, yet extend the expedition to 25 miles.

The use of more travelers will be required to extend the expedition the desired 50 miles. The full

expedition using legs with increasing length of 5 miles will be discussed at a later point. In reaching the

North Pole, there are many possible variations to the complete expedition. These variations, however,

will impact the number of legs that are needed and the number of travelers.

Adding a Second Leg and Additional Travelers to the trip:

In trying to get to the North Pole, it was important to start small and explore the possibilities

when adding a second leg to the expedition. With two legs and one traveler at each leg, it became

possible to extend one person beyond the 10 miles, with the second leg traveling 15 miles from camp.

The first leg needed to travel 5 miles, leave 10 provisions, and return home the 5 miles. The second leg

2 Provisions will refer to resources that the travelers leave behind. (ex. 10 miles worth of supplies = 1 provisions)

2

would travel 5 miles, refill the 5 provisions that had just been spent to bring the total back to 20, travel

out 10 miles and back 10 miles, and then use the remaining 5 provisions waiting on ice (See Illustration

1).

It was interesting to keep with the idea of

one traveler per leg and increase the number of

legs. Because of the limitation of supplies a

traveler can carry, regardless of how many legs

are added to the expedition, if there is only one

traveler per leg, the distance traveled will be

limited to 20 miles. In order to refill to the

maximum of 20 miles and leave enough to get

home, the increasing distance for each leg

became smaller and smaller until the total

distance traveled reached its limit of 20 miles.

This makes sense being that the most one person

can carry at any given time would be 20 miles

worth of supplies (See Table 1, Illustration 2).

3

Illustration 1: 2 travelers - The first person travels 5 miles, leaves 10 provisions behindand returns 5 miles back to camp. The second person travels 5 miles, takes 5 of the 10provisions that were left (now has enough supplies for 20 miles), travels out and back 10miles, picks up the other 5 provisions and returns to camp.

Illustration 2: As the number of legs increase, the number of travelers increases at the same rate. If the limit for travelers is one per leg, the distance traveledeventually maximizes at 20 miles.

Leg (1 travelerper leg)

Distance Traveled IncreasedDistance fromPrevious Leg

Leg (1 travelerper leg)

Distance Traveled Increased Distancefrom Previous Leg

Let x be the leg Let an be the distance for current legFor legs > 1:

1 10.0000 --- 11 19.9902 0.0098

2 15.0000 5.0000 12 19.9951 0.0049

3 17.5000 2.5000 13 19.9976 0.0024

4 18.7500 1.2500 14 19.9988 0.0012

5 19.3750 0.6250 15 19.9994 0.0006

6 19.6875 0.3125 16 19.9997 0.0003

7 19.8438 0.1563 17 19.9998 0.0002

8 19.9219 0.0781 18 19.9999 0.0001

9 19.9609 0.0391 19 20.0000 0

10 19.9805 0.0195 Table 1

Doubling the first-leg travelers allowed for two travelers on the second leg to reach the 15 miles

from base-camp. This is because there are additional supplies left at the drop point, allowing more

4

anan1512 x2

people to travel. It essentially took the idea of one traveler at

each leg and doubled it (See Illustration 3). This idea of

adding more travelers is the key to extending the distance and

eventually reaching the 50 miles needed to get to the North

Pole.

More Legs and More Travelers – MaximumRange Extended:

While adding a second leg to the expedition extended

the distance that a person could travel, it was going to be

necessary to add more legs and more travelers to continue

getting closer to the North Pole. Adding a third and fourth leg to the expedition opened up a number of

variations that would each have differing impacts to the distance traveled and the number of travelers

needed to reach that distance. The distance

traveled depends on the length of the legs. There

are two ways one could work with the length of

the legs: varying the length of the legs throughout

the expedition, or have a constant length added to

each leg. With the varying lengths, there becomes

in infinite number of variations to the expedition

and becomes very difficult to measure and predict

the number of legs needed and the number of

travelers needed to get one person to the North

Pole and back (See Illustration 4 for example of

5

Illustration 4: This illustration does not reveal the number of people that would beneeded to accomplish all of the traveled legs. It merely displays what is meant by legsof varying length. Notice the initial distance traveled at each leg is increasing. This isthe distance of varying length on each leg.

Illustration 3: With the leg lengths of 5 miles, increasing thenumber of people on the first leg to two allows for two people onthe second leg to travel 15 miles from base camp.

varying leg length). The alternative was if the length of the legs remained constant, meaning the

amount traveled and added each leg. With a constant distance, it became much easier to predict how far

the group could travel and how many travelers would be needed (See Illustration 5 for constant leg

length).

Several keys discovered when working withmultiple legs:

1. The key is when stopping at a drop-off site,there be enough supplies to refill completelyback to 20 miles to continue and also to coverthe return home from that point.

2. On the final leg of the expedition it is bestto have 20 provisions when finishing off thatleg. This allows for a maximum of 10 milesand back from the last drop-off point.

3. Varying the lengths of the legs will impactthe number of travelers needed total and ateach leg. The issue that dictates the number oftravelers needed at each leg is the neededsupplies from that check-point to continue toreturn home.

Theoretically, the complete trip to the North Pole and back should be able to be completed in 5

legs, assuming that there is an infinite number of people making the expedition at each leg. The

theoretical is that each traveler would extend their journey as far as possible, leaving the slightest

amount of supplies in the ice. With there being an infinite number of travelers at each leg. The

individual on the next leg would be able to replenish their supplies up to 20 at each transition point.

Each leg could theoretically get infinitely close to 10 miles traveled and returned, therefore, adding 10

miles to the expedition at each leg. If the number of travelers are not an issue, three legs should be able

to get infinitely close to 30 miles and fours legs infinitely close to 40 miles. This suggests that,

6

Illustration 5: This illustration does not reveal the number of people thatwould be needed to accomplish all of the traveled legs. It merely displayswhat is meant by legs of constant length. Notice the initial distance traveled ateach leg remains the same and the increase at each leg is a constant length of5 miles.

theoretically, the expedition could be completed in five

legs if there were an infinite number of people at camp

(See Illustration 6).

The best way to illustrate the maximum distance

traveled, without working in the theoretical, for three

and four legs would be to travel out 9 miles, leave 2

provisions, and return home 9 miles. The next leg would

then do the same, and so on and so forth. The key is

determining how many travelers at leg one, leg two, and

leg three would be needed to continue all the ensuing

legs (See Illustration 7). The ability to determine the

number of travelers for an expedition will be discussed at a later point.

7

Illustration 6: This illustration displays how the increased distance of eachleg would take place. The number of travelers needed to accomplish thecomplete trip of the North Pole that takes place int he 5th leg is not show. Inthis theoretical, there would be an infinite number of travelers required at theprevious legs prior to that final person completing the 5th leg and reachingthe North Pole.

Illustration 7: Next to each leg is the number of people needed for that leg in order for the final traveler at the 6th leg to reach the North Pole. Being that thedistance increase of each leg is 9 miles, there are only 2 provisions left at the end of each leg. With so few provisions left, there is a need for many more travelersto allow for future legs to continue. (For example, at the end of the 1st leg there are 2 provisions left on ice. The 2nd leg travels out 9 miles to that drop point. Inorder to load up to 20 miles worth and travel home the final 9 miles from that same point, there needs to be 18 provisions left. This 18 provisions would requirenine people at the 1st leg to satisfy just one traveler at the 2nd leg. This growth continues throughout the rest of the expedition, explaining why so many peopleare needed.)

Reaching the North Pole:

Reaching the North Pole was possible, but difficult to predict or measure if there was not

consistency to the length of the legs. A consistent length needed to be added at each leg, except the

final leg of the expedition. The final leg did not have to be cut short to leave supplies for any following

travelers. Once a traveler had enough supplies to get to the North Pole and return to the check-point,

there was no need for additional legs. Therefore, the final part of the final leg would look different than

all the preceding legs. Using the same number of miles traveled at each leg in trying to get to the North

Pole was important to maintain consistency.

Expedition with Legs of Length 2 miles:

The first expedition worked out started

by traveling 2 miles, leaving 16 provisions, and

returning home 2 miles. Each of the following

legs would add a distance of 2 to the previous

leg, leave supplies, and then return home 2

miles at a time, picking up supplies along the

way (See Illustration 8). In creating a

spreadsheet it became possible to determine

how many travelers would be needed at each

leg to fulfill the future legs. With a leg length

of 2 miles, a traveler would leave 16 provisions

in the ice. This would both replenish and give

supplies for a return trip home to 4 additional

travelers. Each traveler would need to 2 miles

8

Illustration 8: This illustration does not reveal the number of people that would be neededat each leg to accomplish all of the traveled legs. It merely displays legs with an increasingincrement of 2 miles per leg. Notice the initial distance traveled at each leg remains thesame and the increase at each leg is a constant length of 2 miles. The final leg (21st leg)reveals all the previous increases of 2 miles, contributed by all the previous legs, alongwith the final 10 miles traveled at the end of the leg, reaching the North Pole.

worth to refill back to 20, and then take 2 more miles worth on the return trip home. With a leg length

of 2 miles, it turned out that there would be drop-off points throughout the expedition that would leave

unused supplies in the ice (See table 2 for traveler breakdown).

Traveling to the North Pole using increasing leg lengths of 2 miles: (Table 2)

Miles

traveled

Total

travelers

2 Leg 1 1

4 Leg 2 Leg 1 2

6 Leg 3 Leg 2 Leg 1 3

8 Leg 4 Leg 3 Leg 2 Leg 1 4

10 Leg 5 Leg 4 Leg 3 Leg 2 Leg 1 5

12 Leg 6 Leg 5 Leg 4 Leg 3 Leg 2 2 *Leg1

7

14 Leg 7 Leg 6 Leg 5 Leg 4 Leg 3 2 *Leg2

2 *Leg1

9

16 Leg 8 Leg 7 Leg 6 Leg 5 Leg 4 2 *Leg3

2 *Leg2

3 *Leg1

12

18 Leg 9 Leg 8 Leg 7 Leg 6 Leg 5 2 *Leg4

2 *Leg3

3 *Leg2

3 *Leg1

15

20 Leg10

Leg 9 Leg 8 Leg 7 Leg 6 2 *Leg5

2 *Leg4

3 *Leg3

3 *Leg2

4 *Leg1

19

22 Leg11

Leg10

Leg 9 Leg 8 Leg 7 2 *Leg6

2 *Leg5

3 *Leg4

3 *Leg3

4 *Leg2

5 *Leg1

24

24 Leg12

Leg11

Leg10

Leg 9 Leg 8 2 *Leg7

2 *Leg6

3 *Leg5

3 *Leg4

4 *Leg3

5 *Leg2

6 *Leg1

30

26 Leg13

Leg12

Leg11

Leg10

Leg 9 2 *Leg8

2 *Leg7

3 *Leg6

3 *Leg5

4 *Leg4

5 *Leg3

6 *Leg2

8 *Leg1

38

28 Leg14

Leg13

Leg12

Leg11

Leg10

2 *Leg9

2 *Leg8

3 *Leg7

3 *Leg6

4 *Leg5

5 *Leg4

6 *Leg3

8 *Leg2

10 *Leg1

48

30 Leg15

Leg14

Leg13

Leg12

Leg11

2 *Leg10

2 *Leg9

3 *Leg8

3 *Leg7

4 *Leg6

5 *Leg5

6 *Leg4

8 *Leg3

10 *Leg2

12 *Leg1

60

32 Leg16

Leg15

Leg14

Leg13

Leg12

2 *Leg11

2 *Leg10

3 *Leg9

3 *Leg8

4 *Leg7

5 *Leg6

6 *Leg5

8 *Leg4

10 *Leg3

12 *Leg2

15 *Leg1

75

34 Leg17

Leg16

Leg15

Leg14

Leg13

2 *Leg12

2 *Leg11

3 *Leg10

3 *Leg9

4 *Leg8

5 *Leg7

6 *Leg6

8 *Leg5

10 *Leg4

12 *Leg3

15 *Leg2

19 *Leg1

94

36 Leg18

Leg17

Leg16

Leg15

Leg14

2 *Leg13

2 *Leg12

3 *Leg11

3 *Leg10

4 *Leg9

5 *Leg8

6 *Leg7

8 *Leg6

10 *Leg5

12 *Leg4

15 *Leg3

19 *Leg2

24 *Leg1

118

38 Leg19

Leg18

Leg17

Leg16

Leg15

2 *Leg14

2 *Leg13

3 *Leg12

3 *Leg11

4 *Leg10

5 *Leg9

6 *Leg8

8 *Leg7

10 *Leg6

12 *Leg5

15 *Leg4

19 *Leg3

24 *Leg2

30 *Leg1

148

40 Leg20

Leg19

Leg18

Leg17

Leg16

2 *Leg15

2 *Leg14

3 *Leg13

3 *Leg12

4 *Leg11

5 *Leg10

6 *Leg9

8 *Leg8

10 *Leg7

12 *Leg6

15 *Leg5

19 *Leg4

24 *Leg3

30 *Leg2

37 *Leg1

185

50 Leg21

Leg20

Leg19

Leg18

Leg17

2 *Leg16

2 *Leg15

3 *Leg14

3 *Leg13

4 *Leg12

5 *Leg11

6 *Leg10

8 *Leg9

10 *Leg8

12 *Leg7

15 *Leg6

19 *Leg5

24 *Leg4

30 *Leg3

37 *Leg2

47 *Leg1

232

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Expedition with Legs of Length 5 miles:

Working with legs with length of 5 miles proved to be easier than working with a length of 2

miles. Unlike a leg of 2 miles, the leg of 5 miles did not waste any materials in the ice. The traveling of

5 miles, leaving 10 provisions, and returning 5 miles maximized the distance traveled while using all

the supplies and not leaving anything in the ice. The deposit of 10 miles worth of supplies would refill

and give return trip home for one traveler at the next leg. There would be no materials left in the ice.

Because each traveler would allow for one additional traveler at the next leg, the increase in needed

travelers increased exponentially by 2x-1. (See Illustration and Table 3)

10

Illustration 9: Next to each leg is the number of people needed for that leg in order for the final traveler at the 9th leg to reach the North Pole. Being that the distanceincrease of each leg is 5 miles, there are 10 provisions left at the end of each leg. With those provisions left, there are enough provisions for one person at the following legto refill up to 20 provisions and have 5 provisions left on ice for the return trip. At the end of each leg, the 10 provisions that are left are color coded to the part of thefollowing leg that the provisions satisfy.

Traveling to the North Pole using increasing leg length of 5 miles: (Table 3)

Milestraveled

Total travelers:2x-1

(where x is the leg)

5 Leg 1 1

10 Leg 2 Leg 1 2

15 Leg 3 Leg 2 2 (Leg 1) 4

20 Leg 4 Leg 3 2 (Leg 2) 4 (Leg 1) 8

25 Leg 5 Leg 4 2 (Leg 3) 4 (Leg 2) 8 (Leg 1) 16

30 Leg 6 Leg 5 2 (Leg 4) 4 (Leg 3) 8 (Leg 2) 16 (Leg 1) 32

35 Leg 7 Leg 6 2 (Leg 5) 4 (Leg 4) 8 (Leg 3) 16 (Leg 2) 32 (Leg 1) 64

40 Leg 8 Leg 7 2 (Leg 6) 4 (Leg 5) 8 (Leg 4) 16 (Leg 3) 32 (Leg 2) 64 (Leg 1) 128

50 Leg 9 Leg 8 2 (Leg 7) 4 (Leg 6) 8 (Leg 5) 16 (Leg 4) 32 (Leg 3) 64 (Leg 2) 128 (Leg 1) 256

Expedition with Legs of Any Length:

Regardless of what length of legs were used for the expedition, the key was determining how

many people the supplies left behind would cover. Working backwards, it became possible to, first,

determine the number of legs that would be needed based on the length of the legs and then the number

of people needed. The expedition with length of 2 miles per leg required 21 legs and the expedition

with length of 5 miles per leg required 9 legs. Once it was determined how many legs would be

required to reach the North Pole, the number of travelers at each leg could also be attained based on the

amount of supplies left at each leg and how many of the following travelers that would cover.

Throughout the journey, travelers will be leaving supplies in the ice for the following journeymen. The

final leg, however, will get to travel further at the end because that traveler does not have to leave any

supplies. This is accounted for with the piece that is subtracted from 50.

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- Let x equal the distance traveled for each leg.

- Therefore, (20-2x/2) represents the amount subtracted from 50due to the fact the final leg does not need to leave supplies.

Distance of Leg(in miles)

Number of Legs Neededto Reach 50 miles

(would have to go up to the nextnumber when not even number of

legs)

Number of legs needed for 50 miles =

50202 x2

x

1 412 213 14.33 15

4 115 96 7.67 8

7 6.71 7

8 69 5.44 6

½ 81

Once it is determined how many legs will be needed, the number of travelers at each leg can be

attained. It is best to work backwards, starting with the final leg. Knowing that there is only a need for

one of the final leg, it is necessary to figure out how many of each of the preceding legs is needed to

get all of the travelers that come after. There is also only a need for one traveler for the leg immediately

preceding the final leg. The key is to determine how many people are needed for all the other legs. The

following equation breaks down how many travelers would be needed for each leg. Notice the

inequality is set greater than, or equal to zero. This greater than, or equal to zero, represents enough

provisions for those travelers that will follow. A negative number would mean there weren't enough

provisions in the ice at that leg to satisfy future travelers. While it is important that there be enough

provisions left on ice for future travelers, it is ideal to complete the trip with the greatest level of

efficiency. To accomplish this, the expedition wants to use the smallest number of travelers at each leg

that still maintains that positive number and, therefore, enough materials at the drop-off points. The

following equation, when used with the correct number of legs, will give the most efficient number of

travelers needed for the expedition. The equation will calculate how many people the provisions at the

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drop-off point for the current leg will satisfy. By changing the number of travelers for the given leg

(an), it becomes possible to find the ideal number of travelers for each leg, while ensuring there are

enough provisions throughout the expedition.

- Let x equal the distance traveled for each leg.

- Therefore, (20-2x/2x) represents the number of following travelers the current leg will providesupplies for.

- Let an be the number of travelers needed for the given leg.

- Let an+y be the number of travelers in last leg in the series.

0an202x

2xan1an2 a n3 ... an y

With the use of this equation, it becomes possible to calculate all the travelers needed at each leg to

ensure all travelers have enough provisions to continue their journey, and eventually, reach the North

Pole. The following table displays how this equation would work for an expedition with legs of 2 miles

and an expedition with legs of 5 miles.

Legs Let x equal the distance

traveled for each leg

Number oftravelers at each

leg

Provisionsremaining 0 # <

(20-2x/2x)

Legs Let x equal the distance

traveled for each leg

Number oftravelers at each

leg

Provisionsremaining 0 # <

(20-2x/2x)

x = 2 an (20-2x/2x) = 4 x = 5 an (20-2x/2x) = 1

21 1 9 1

20 1 3 8 1 0

19 1 2 7 2 0

18 1 1 6 4 0

17 1 0 5 8 0

16 2 3 4 16 0

15 2 1 3 32 0

14 3 3 2 64 0

13 3 0 1 128 0

12 4 1

11 5 1

10 6 0

13

9 8 2

8 10 2

7 12 0

6 15 0

5 19 1

4 24 2

3 30 2

2 37 0

1 47 3

Total Travelers 232 Total Travelers 256

If one knows the length of the legs that will be used to complete the journey, they can determine the

number of legs that will be needed to travel the 50 miles and reach the North Pole. With the length of

the legs and the number of legs known, it can then be determined how many travelers will be needed at

each leg and, therefore, the complete journey. The following shows the number of travelers that will be

needed for each of the previously discussed expeditions of various lengths.

Length of Legs Number of Legs needfor North Pole

Total Number ofTravelers

½ 81 672

1 41 376

2 21 2323 14.33 15 332

4 11 265

5 9 2566 7.67 8 783

7 6.71 7 1747

8 6 3125

9 5.44 6 10000

After analyzing the number of travelers that would be needed for each of the various expeditions, there

appears to be a “sweet spot” between the leg distance of 2 miles and 5 miles. A distance of 3 miles per

leg is a little bit out of the pattern because it requires the 14.33 legs to be rounded up to 15 to complete

14

the journey. This additional leg requires more travelers throughout the journey. The expedition that is

made up of 2-mile legs requires the fewest number of travelers to complete. It does, however, waste

provisions that are left unused at drop-off points. The expedition that uses all the provisions along the

way and still uses a low number of travelers was the trip comprised of 5-mile legs.

Extending Beyond the North Pole:

Let's suppose that there are 1000 people at the base camp. Using the ideas in the previous

section, it can determined how far beyond the 50 miles each of the expeditions could carry on. Adding

legs to the expedition and then extending the number of travelers needed to ensure enough provisions,

it is possible to determine the number of travelers for any expedition. The following illustrates what

happens with an expedition with 2-mile legs. It appears that 26 legs are possible to complete with 888

travelers. The 27th leg brings the needed number of travelers up to 1110, beyond the 1000 people. 26

legs on the 2-mile expedition would travel out 60 miles and back. The 5-mile legs can get to the same

distance of 60 miles and back, but need 1024 travelers to do it. This requires more than the limit 1000

people that might be in camp.

As the trip continues beyond the 50 miles, the separation of the number of travelers needed for

each of the various expeditions grows. Regardless of the length of the legs, it is possible for an infinite

number of travelers to continue the voyage forever. It just requires a lot of people to do it. The

following breaks down how many travelers are needed for a voyage of 100 miles for several of the leg-

lengths:

Length of Legs Number of Legs needfor 100 miles

Total Number ofTravelers

2 46 61,6943 31 100,1905 19 262,1449 11 10,000,000,000

15

The trip of 50 miles to the North Pole showed that the trip could be done with various leg-

lengths, and that these variations to the expedition would result in different numbers of travelers

needed. If the voyage that the people will engage in is extended beyond that 50 miles, the differences in

the number of people required grows very rapidly and the separation between the legs and the number

of people required at each leg grows.

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