Henry Ernest Dudeney/Modern Puzzles/50 - Exploring Mount Neverest/Solution
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Modern Puzzles by Henry Ernest Dudeney: $50$
- Exploring Mount Neverest
- Professor Walkingholme, one of the exploring party, was allotted the special task of making a complete circuit of the base of the mountain at a certain level.
- The circuit was exactly $100$ miles in length and he had to do it all alone on foot.
- He could walk $20$ miles a day, but he could only carry rations for $2$ days at a time,
- the rations for each day being packed in sealed boxes for convenience in dumping.
- He walked his full $20$ miles every day and consumed $1$ day's ration as he walked.
- What is the shortest time in which he could complete the circuit?
Solution
- $23 \tfrac 1 2$ days.
Working
| \(\text {(1)}: \quad\) | \(\ds \) | \(\) | \(\ds \) | Dump $5$ rations at $90$-mile point and return to base | \(\quad\) ($5$ days) | ||||||||||
| \(\text {(2)}: \quad\) | \(\ds \) | \(\) | \(\ds \) | Dump $1$ at $85$ and return to $90$ | \(\quad\) ($1$ day) | ||||||||||
| \(\text {(3)}: \quad\) | \(\ds \) | \(\) | \(\ds \) | Dump $1$ at $80$ and return to $90$ | \(\quad\) ($1$ day) | ||||||||||
| \(\text {(4)}: \quad\) | \(\ds \) | \(\) | \(\ds \) | Dump $1$ at $80$, return to $85$, pick up $1$, dump at $80$ | \(\quad\) ($1$ day) | ||||||||||
| \(\text {(5)}: \quad\) | \(\ds \) | \(\) | \(\ds \) | Dump $1$ at $70$, return to $80$ | \(\quad\) ($1$ day) | ||||||||||
| \(\text {(6)}: \quad\) | \(\ds \) | \(\) | \(\ds \) | Return to base | \(\quad\) ($1$ day) | ||||||||||
| We now have $1$ at $70$ and $1$ at $90$. | |||||||||||||||
| \(\text {(7)}: \quad\) | \(\ds \) | \(\) | \(\ds \) | Dump $1$ at $5$ and return to base | \(\quad\) ($1$ day) | ||||||||||
| Note that he has only walked $10$ miles on the above day, but he can always walk to $10$ and back, dropping off the ration at $5$ on the way. | |||||||||||||||
| \(\text {(8)}: \quad\) | \(\ds \) | \(\) | \(\ds \) | Dump $4$ at $10$ and return to base | \(\quad\) ($4$ days) | ||||||||||
| \(\text {(9)}: \quad\) | \(\ds \) | \(\) | \(\ds \) | Dump $1$ at $10$ and return to $5$, pick up $1$ and go to $10$ | \(\quad\) ($1$ day) | ||||||||||
| \(\text {(10)}: \quad\) | \(\ds \) | \(\) | \(\ds \) | Dump $2$ at $20$ and return to $10$ | \(\quad\) ($2$ days) | ||||||||||
| \(\text {(11)}: \quad\) | \(\ds \) | \(\) | \(\ds \) | Dump $1$ at $25$ and return to $20$ | \(\quad\) ($1$ day) | ||||||||||
| \(\text {(12)}: \quad\) | \(\ds \) | \(\) | \(\ds \) | Dump $1$ at $30$, return to $25$, pick up $1$ and return to $30$ | \(\quad\) ($1$ day) | ||||||||||
| \(\text {(13)}: \quad\) | \(\ds \) | \(\) | \(\ds \) | March to $70$ | \(\quad\) ($2$ days) | ||||||||||
| \(\text {(14)}: \quad\) | \(\ds \) | \(\) | \(\ds \) | March to base | \(\quad\) ($1 \tfrac 1 2$ days) | ||||||||||
| \(\ds \) | \(\) | \(\ds \) | Total: | \(\quad\) $23 \tfrac 1 2$ days | |||||||||||
If the route is a straight march across a desert, that is, without being able to start at the end and work backwards, the minimum time is $86$ days, as follows:
| \(\text {(1)}: \quad\) | \(\ds \) | \(\) | \(\ds \) | Dump $42$ at $10$ and return to base | \(\quad\) ($42$ days) | |||||||||
| \(\text {(2)}: \quad\) | \(\ds \) | \(\) | \(\ds \) | Dump $1$ at $15$ and return to $10$ | \(\quad\) ($1$ day) | |||||||||
| \(\text {(3)}: \quad\) | \(\ds \) | \(\) | \(\ds \) | Dump $20$ at $20$ and return to $10$ | \(\quad\) ($20$ days) | |||||||||
| \(\text {(4)}: \quad\) | \(\ds \) | \(\) | \(\ds \) | Dump $1$ at $20$, return to $15$, pick up $1$, dump at $20$ | \(\quad\) ($1$ day) | |||||||||
| \(\text {(5)}: \quad\) | \(\ds \) | \(\) | \(\ds \) | Dump $10$ at $30$, return to $20$ | \(\quad\) ($10$ days) | |||||||||
| \(\text {(6)}: \quad\) | \(\ds \) | \(\) | \(\ds \) | Dump $1$ at $35$, return to $30$ | \(\quad\) ($1$ day) | |||||||||
| \(\text {(7)}: \quad\) | \(\ds \) | \(\) | \(\ds \) | Dump $4$ at $40$ and return to $30$ | \(\quad\) ($4$ days) | |||||||||
| \(\text {(8)}: \quad\) | \(\ds \) | \(\) | \(\ds \) | Dump $1$ at $40$ and return to $35$, pick up $1$ and dump at $40$ | \(\quad\) ($1$ day) | |||||||||
| \(\text {(9)}: \quad\) | \(\ds \) | \(\) | \(\ds \) | Dump $2$ at $50$ and return to $40$ | \(\quad\) ($2$ days) | |||||||||
| \(\text {(10)}: \quad\) | \(\ds \) | \(\) | \(\ds \) | Dump $1$ at $55$ and return to $50$ | \(\quad\) ($1$ day) | |||||||||
| \(\text {(11)}: \quad\) | \(\ds \) | \(\) | \(\ds \) | Dump $1$ at $60$ and return to $55$, pick up $1$ and dump at $60$ | \(\quad\) ($1$ day) | |||||||||
| \(\text {(12)}: \quad\) | \(\ds \) | \(\) | \(\ds \) | March to $100$ and the end of the desert | \(\quad\) ($2$ days) | |||||||||
| \(\ds \) | \(\) | \(\ds \) | Total: | \(\quad\) $86$ days |
Sources
- 1926: Henry Ernest Dudeney: Modern Puzzles ... (previous) ... (next): Solutions: $50$. -- Exploring Mount Neverest
- 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $77$. Exploring Mount Neverest