Seventeen Camels and Where They Can Take You

“Oh, it’s just a trick thing.” — Ben Ames Williams, Coconuts

Below are six mathematical puzzles. While they appear unrelated at first glance, they share a hidden commonality. I will first present the problems, then offer a hint regarding their shared nature, and finally provide solutions that utilize a specific, elegant "trick."

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Puzzle 1: The Merchant's Legacy

A wealthy man passed away, leaving behind 17 camels to be split among his three heirs according to a specific will:

Heir	Share
Eldest	$1/2$ of the total
Second	$1/3$ of the total
Youngest	$1/9$ of the total

~~The most literal approach would be to slaughter the animals to achieve exact fractions: 8.5 camels for the first, 5.66 for the second, and 1.88 for the third, leaving a tiny scrap for the vultures.~~

While the heirs were stuck in a deadlock, a traveling trader arrived. After hearing the dilemma, he offered a single, simple suggestion that resolved the conflict perfectly.

Puzzle 2: The Coconut Conundrum

Five sailors and a monkey were stranded on a desert island. After a day of gathering coconuts, they piled them up and went to sleep.

Sailor 1 woke up, divided the pile into five equal parts, found one coconut left over (which he gave to the monkey), and hid one of the five piles.
Sailor 2 did the exact same thing with the remaining pile: divided by five, gave the remainder to the monkey, and hid one pile.
This process repeated for Sailors 3, 4, and 5.

In the morning, the remaining coconuts were divided equally among the five sailors with no remainder.

The Goal: Find the smallest possible number of original coconuts.

Warm-up challenge: Find the smallest positive integer $x$ such that: x mod 3 = 2 x mod 5 = 4 x mod 7 = 6

In $\LaTeX$ : $x \equiv 2 \pmod 3, \quad x \equiv 4 \pmod 5, \quad x \equiv 6 \pmod 7$

Puzzle 3: The Discrete Math Dilemma

Imagine you are in a Discrete Mathematics course. You recall a formula regarding graph theory.

Key Definitions:

Graph: A set of vertices (dots) connected by edges (lines).
Connected: A graph where every vertex is reachable from every other vertex.
Cycle: A path that starts and ends at the same vertex without repeating edges.
Tree: A connected graph with no cycles.

The formula you remember is $E = V - 1$ (where $E$ is edges and $V$ is vertices).

Graph Example The image above shows a graph that is neither connected nor a tree (it contains a cycle).

Tree Example This image shows a tree with 10 vertices and 9 edges.

The Exam Question: A forest is a graph with no cycles, but it doesn't have to be connected (it's essentially a collection of trees). If a graph $G$ is a forest consisting of exactly two trees, how many edges does it have?

Puzzle 4: The Counterfeit Coin

You are presented with 4 coins. One is counterfeit and has a different weight (it could be heavier or lighter than the others).

The Question: Can you identify the fake coin and determine its weight property (heavy/light) using only two weighings on a balance scale?

If this is impossible, what single additional object could you add to the mix to make it possible?

Puzzle 5: The Jack of All Trades

You have a standard, randomly shuffled deck of 52 cards. You begin dealing cards one by one until you encounter one of the 4 jacks, at which point you stop.

The Question: On average, how many cards will you have dealt?

Cards

Puzzle 6: The Grand Swap

Consider a deck of $N$ cards numbered $1$ through $N$ . A "Grand Swap" is an $N$ -step process:

Swap the cards above the 1 with the cards below the 1.
Swap the cards above the 2 with the cards below the 2.
...Repeat until you swap the cards above and below $N$ .

Note: If a "part" is empty, the swap simply moves the top card to the bottom or vice versa.

Example with $N=4$ (Starting Order: 1, 2, 4, 3):

Initial: [1, 2, 4, 3]
Step 1 (Swap around 1): [2, 4, 3, 1]
Step 2 (Swap around 2): [4, 3, 1, 2]
Step 3 (Swap around 3): [1, 2, 3, 4]
Step 4 (Swap around 4): [4, 1, 2, 3]

Swap 1 Swap 2

Final Notes

These puzzles may seem to span different worlds—from probability and graph theory to modular arithmetic and logic—but they all yield to the same conceptual "trick."

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