The Shipwreck Puzzle – A Fun Linear Math Challenge

The shipwreck puzzle is a fun challenge that tests your fundamental understanding of rates and linear algebra. The puzzle begins when two ships get wrecked by a meteor strike and start sinking at the same time. Both ships start sinking at sea level, which can be considered ‘h’.

On the one hand, it would take Ship-1 ten hours to reach the bottom of the sea. On the other hand, it would take Ship-2 just five hours to reach the bottom of the sea owing to heavier damage. The seafloor can be considered to be at a height of zero metres. Now, given this premise, the challenge is as follows:

How long (in time) after the ships start sinking, will Ship-2 be at half the height of Ship-1 before Ship-2 hits the seafloor?

Both ships start sinking at the same time and are sinking at a constant linear rate. That is pretty much it. This is a fairly straightforward challenge, but it requires some basic analytical thinking, which is a fun way to practice math.

Spoiler Alert: From this point onwards, I will directly be discussing the solution to the puzzle. So, if you wish to give this puzzle a crack on your own, now is the time. Later on, you may choose to revisit the essay and compare your approach/solution.

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Establishing Slopes and Lines

The first important clue that we have is that the ships start from the same height and are sinking at a constant linear rate. It just happens to be the case that Ship-2 is sinking at twice the rate of Ship-1.

If we were to visualize this problem as a graph with the height of the ships on the Y-axis and the time taken on the X-axis, it would present itself as follows:

Because both ships sink at a constant linear rate, we obtain straight lines that flow from ‘h’ on the Y-axis towards the X-axis at different rates. The line for Ship-1 has a slope of m₁ = -(h/10) and the line for Ship-2 has a slope of m₂ = -(h/5).

Solving the Shipwreck Puzzle

Now that we have the slopes of both these lines, we need only to plug them into the respective line equations. The general equation for a line is given by:

Y = (m*x) + c, where m is the slope and c is the y-intercept.

In our case, let y1 and y₂ be the instantaneous height for Ships-1 and 2 respectively. Consequently, we obtain the following two line-equations:

y₁ = (m₁*x) + h = -(h/10)*x + h

y₂ = (m₂*x) + h = -(h/5)*x + h

We have now, everything we need to solve the main puzzle.

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The Solution to the Shipwreck Puzzle

Just as a reminder, the following is the puzzle’s challenge:

How long (in time) after the ships start sinking, will Ship-2 be at half the height of Ship-1 before Ship-2 hits the seafloor?

Considering that y₁ and y₂ are the instantaneous heights of Ships-1 and 2 respectively, we have to satisfy the following expression:

y₂ = ½ *y₁ (because Ship-2 needs to be at half the height of Ship-1)

y₁ = 2 *y₂

From this point onwards we could just plug in the respective expressions for y₁ and y2 and solve for ‘x’ (time) using algebra.

The fact that there is no more ‘h’ in the expression just goes to show that the puzzle is independent of the initial height from which the ships started to sink. Proceeding onward, we get the following result:

There we go. Ship-2 would be at half the height of Ship-1 after 10/3 hours or 3 hours and 20 minutes.

Final Remarks

Asa final thought, I just wish to mention that this is not the only approach to solving the problem, but the solution will have to be the same.

There are several shorter approaches to this puzzle, but I wanted to present what I felt was the most inclusive and intuitive approach. This way, everyone gets to enjoy the puzzle. On that note, I hope you enjoyed the challenge!

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Credit: My work on this essay was inspired by the work done by Presh Talkwalkar.

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Further reading that might interest you:  Can You Really Solve This Third-Grade Math Puzzle? and How Much String Would You Need To Wrap Around The Earth?