A trekking trail rises steadily from the Base camp to the Top camp. There are six intermediate camps, C1 to C6, and the trail never descends between consecutive camps. The distance of each camp from Base (along the trail) and its altitude are given below.
| Camp | Distance from Base (km) | Altitude (m) |
|---|---|---|
| Base | 0 | 2000 |
| C1 | 5 | 2400 |
| C2 | 11 | 2900 |
| C3 | 14 | 3300 |
| C4 | 20 | 3600 |
| C5 | 27 | 4100 |
| C6 | 31 | 4600 |
| Top | 36 | 4900 |
Unless a question says otherwise, trekkers follow these rules on the ascent:
- Each trekking day starts at one camp and ends at a later camp; nights are spent only at camps.
- The distance walked in a day cannot exceed 12 km.
- The altitude gained in a day (altitude of the camp where the day ends minus the altitude of the camp where it starts) cannot exceed 900 m.
- The first time a trekker spends a night at a camp above 4000 m (other than Top), he or she must spend the whole of the next day resting at that camp. This rest day counts as a day of the trek.
The ascent is complete on arrival at Top.
| Camp | Distance from Base (km) | Altitude (m) |
|---|---|---|
| Base | 0 | 2000 |
| C1 | 5 | 2400 |
| C2 | 11 | 2900 |
| C3 | 14 | 3300 |
| C4 | 20 | 3600 |
| C5 | 27 | 4100 |
| C6 | 31 | 4600 |
| Top | 36 | 4900 |
Unless a question says otherwise, trekkers follow these rules on the ascent:
- Each trekking day starts at one camp and ends at a later camp; nights are spent only at camps.
- The distance walked in a day cannot exceed 12 km.
- The altitude gained in a day (altitude of the camp where the day ends minus the altitude of the camp where it starts) cannot exceed 900 m.
- The first time a trekker spends a night at a camp above 4000 m (other than Top), he or she must spend the whole of the next day resting at that camp. This rest day counts as a day of the trek.
The ascent is complete on arrival at Top.
Question 1
Numerical answer
What is the minimum number of days needed to complete the ascent from Base to Top?
Show answer and solution
Answer: 5
Legal single-day moves (≤ 12 km and ≤ 900 m gain):
- Base → C1, C2
- C1 → C2, C3
- C2 → C3, C4
- C3 → C4
- C4 → C5 (C4 → C6 gains 1000 m)
- C5 → C6, Top
- C6 → Top
C4 must be reached, and from C4 the only move is to C5 (above 4000 m, so a rest day follows). From C5 one can go straight to Top (9 km, 800 m).
Reaching C4 takes at least 2 days (Base → C2 → C4, the only 2-day way, since Base cannot reach C3 or C4 in a day). So the fastest ascent is Base → C2 → C4 → C5 → (rest) → Top = 4 walking days + 1 rest day = 5 days, and it is the only 5-day plan.
- Base → C1, C2
- C1 → C2, C3
- C2 → C3, C4
- C3 → C4
- C4 → C5 (C4 → C6 gains 1000 m)
- C5 → C6, Top
- C6 → Top
C4 must be reached, and from C4 the only move is to C5 (above 4000 m, so a rest day follows). From C5 one can go straight to Top (9 km, 800 m).
Reaching C4 takes at least 2 days (Base → C2 → C4, the only 2-day way, since Base cannot reach C3 or C4 in a day). So the fastest ascent is Base → C2 → C4 → C5 → (rest) → Top = 4 walking days + 1 rest day = 5 days, and it is the only 5-day plan.
Question 2
Single correct option
In the fastest possible ascent, at which of the following camps does the trekker NOT spend a night?
- A. C2
- B. C4
- C. C5
- D. C3
Show answer and solution
Answer: (D) C3
The unique fastest ascent is Base → C2 → C4 → C5 (night, then a rest day and a second night) → Top.
Nights are spent at C2, C4 and C5. C3 is never a night halt.
Why C3 cannot help: Base → C3 is 14 km (too far), and going through C1 → C3 costs 2 days to reach C3, which is no better than reaching C4 in 2 days.
Nights are spent at C2, C4 and C5. C3 is never a night halt.
Why C3 cannot help: Base → C3 is 14 km (too far), and going through C1 → C3 costs 2 days to reach C3, which is no better than reaching C4 in 2 days.
Question 3
Numerical answer
Suppose the daily altitude-gain limit is raised from 900 m to 1200 m, with all other rules unchanged. How many different itineraries (sequences of night halts) complete the ascent in the minimum possible number of days?
Show answer and solution
Answer: 2
With a 1200 m limit, the new useful move is C4 → C6 (11 km, 1000 m). C4 → Top is still 16 km, and Base still cannot reach C3 or C4 in a day (14 km and 20 km).
The minimum is still 5 days, and these plans achieve it:
1. Base → C2 → C4 → C5 → (rest at C5) → Top
2. Base → C2 → C4 → C6 → (rest at C6, the first night above 4000 m) → Top (5 km, 300 m)
No 4-day plan exists: reaching C4 needs 2 days, the camp after C4 is above 4000 m (so a rest day follows), and Top is not reachable from C4 in a day. Answer: 2.
The minimum is still 5 days, and these plans achieve it:
1. Base → C2 → C4 → C5 → (rest at C5) → Top
2. Base → C2 → C4 → C6 → (rest at C6, the first night above 4000 m) → Top (5 km, 300 m)
No 4-day plan exists: reaching C4 needs 2 days, the camp after C4 is above 4000 m (so a rest day follows), and Top is not reachable from C4 in a day. Answer: 2.
Question 4
Single correct option
Suppose instead that the daily distance limit is raised to 15 km, while the altitude-gain limit stays at 900 m and the rest-day rule is unchanged. How many different itineraries complete the ascent in the minimum possible number of days?
- A. 2
- B. 3
- C. 4
- D. 1
Show answer and solution
Answer: (B) 3
New moves under 15 km: Base → C3 is 14 km but gains 1300 m (not allowed). C1 → C3 (9 km, 900 m), C3 → C5 (13 km, 800 m) and C2 → C3 now matter. C4 → Top (16 km) is still too far.
The minimum is still 5 days (3 walking days to a camp above 4000 m, a rest day, then Top). The 5-day plans are:
1. Base → C2 → C4 → C5 → rest → Top
2. Base → C1 → C3 → C5 → rest → Top
3. Base → C2 → C3 → C5 → rest → Top
A 4-day plan would need to reach C5 or C6 in 2 days, which is impossible (the 2-day reach is at most C4). Answer: 3.
The minimum is still 5 days (3 walking days to a camp above 4000 m, a rest day, then Top). The 5-day plans are:
1. Base → C2 → C4 → C5 → rest → Top
2. Base → C1 → C3 → C5 → rest → Top
3. Base → C2 → C3 → C5 → rest → Top
A 4-day plan would need to reach C5 or C6 in 2 days, which is impossible (the 2-day reach is at most C4). Answer: 3.
Question 5
Single correct option
On the way down, the altitude and rest-day rules do not apply, but a trekker can walk at most 18 km a day and must still halt only at camps. If a trekker makes the fastest possible ascent under the original rules and then the fastest possible descent, how many days does the round trip take?
- A. 6
- B. 7
- C. 8
- D. 9
Show answer and solution
Answer: (C) 8
Ascent: 5 days (see the earlier solution).
Descent: the trail is 36 km. Two days would need a halt at exactly 18 km from Top (distance 18 from Base), and there is no camp there. Three days is possible, for example Top → C4 (16 km) → C1 (15 km) → Base (5 km).
Round trip = 5 + 3 = 8 days.
Trap: ⌈36/18⌉ = 2 ignores that halts must be at camps.
Descent: the trail is 36 km. Two days would need a halt at exactly 18 km from Top (distance 18 from Base), and there is no camp there. Three days is possible, for example Top → C4 (16 km) → C1 (15 km) → Base (5 km).
Round trip = 5 + 3 = 8 days.
Trap: ⌈36/18⌉ = 2 ignores that halts must be at camps.
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