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Searching for lost cities in ancient numbers | Sonarships

Hunt for ancient cities in ancient bookkeeping records, and play a different kind of Battleship.
Double Helix
CSIRO's magazine for kids
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Maths by Email

News: Searching for lost cities in ancient numbers

Cities came and went in the ancient world. To find the lost ones, we can hunt for descriptions of landmarks in archaeological records. A new method instead uses numbers to narrow down the hiding places of buried settlements.
Nobody knows where the town of Durhumit is any more. Any sign of the place has long vanished, either forgotten and renamed, or covered by erosion and time.
Writing on clay tablets found in what was once a trading hub called Kaneš – now modern day Turkey – mentions the place, but doesn’t exactly give clear directions.
It’s not uncommon for archaeological records to provide a hint or two on the location of lost places in the form of landmarks, but what can you do when you only have a name and a few lists of numbers?
Mathematicians from the US National Bureau of Economic research believe there’s a lot of detail we can pull from what are effectively bookkeeping records.
Consider a passage such as this one:

"From Durhumit until Kaneš I incurred expenses of 5 minas of refined (copper), I spent 3 minas of copper until Wahšušana, I acquired and spent small wares for a value of 4 shekels of silver."

It doesn’t say how to get to Durhumit from Kaneš, but it does say how much a traveller spent. Combined with data from numerous other texts, researchers can work out how often people from Kaneš traded with Durhumit.
Since traders will trade more often with closer cities, and no doubt spend less getting there, it’s possible to build a detailed picture of an area where a lost city is most likely to be found.
In this case, the researchers analysed 12 000 clay tablets to develop an idea of the trading centres surrounding Kaneš some 4000 years ago. Using the locations of 15 known ancient cities, they located the likely spots for 11 more.
Durhumit is still lost. But for any archaeologist looking for the place, they now know roughly where to start their hunt for its ruins.

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If you feel like getting hands-on, there’s plenty to do. We show you how to grow beautiful crystals that taste great, too. And then uncover the secret power that moves our oceans.

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Brainteaser question

Kelly is writing down a sequence of words, which follow a simple pattern:
  • seventh
  • sixth
  • fifth
  • fourth
  • third
Kelly’s next word is not ‘second’. Can you work out what it is?
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Maths activity

Try this: Sonarships

You might have played Battleship, but you’ve probably never played it like this.

You will need

  • A friend to play with
  • 2 copies of the sonarships print out
  • 6 coins
  • 2 pencils
  • 2 drawing compasses
  • 2 rulers
  • 2 screens to hide behind. You could use a large hardcover book or a lid from a board game.

What to do

  1. Place a sonarships print out behind each screen, along with three coins. You get one, and your friend gets the other.
  2. Secretly mark three positions on your own grid by colouring in any three squares. They are your ‘ships’.
  3. Ask your friend to secretly mark three positions on their own grid.
  4. Flip a coin to see who goes first.
  5. The first player calls a grid position, such as "F5" or "V17", and marks it with an X on their own grid.
  6. The responding player measures how far the grid position is from any of their ships in centimetres.
  7. If they are within 5 centimetres, they reply with the word "Ping!" for each ship, followed by the measurement. For example, they might say "Ping! 3 centimetres, Ping! 4.5 centimetres".
  8. The first player uses the compass to draw one or more circles, each with a radius of a measurement around the grid position. They can be confident one of the other person’s ships is on the circumference of each circle (or very close to it!).
  9. The second player then calls a grid position. If successful, they then draw one or more circles on their grid in response.
  10. On the first player’s next turn, they suggest another grid position.
  11. At any moment, a player can say "Torpedo!", followed by a grid number. They then put aside one coin. But be careful – you only get three torpedoes.
  12. If the torpedo grid number matches a ship, the ship is sunk.
  13. The game ends when all torpedoes have been launched.
  14. The winner is the player who sinks the most ships.
  15. With some practice, you should be able to work out a strategy that means you can quickly zoom in on a ship using just a few pings.
Enjoy this activity? Visit for more.
Someone is colouring in a square on some graph paper.
Secretly mark three positions on your own grid by colouring in any three squares. They are your ‘ships’.
Someone is drawing a circle on the graph paper using a compass.
Use the compass to draw a circle with the radius that was called out.
Three circles meet near the same point.
When several circles meet at the same point, you might have found one of your opponent's ships!

What’s happening?

With each successful ping, you are given a key bit of information on a hidden sub’s position – how far away it is. If you’re lucky, this could reduce your chances of finding it to just eight squares. If not, it could be on one of 20 or more squares.
In either case, to narrow it down further, you need more circles. The second circle should overlap the first. If both circles overlap in one place, you can consider yourself lucky – that’s your target. More likely, the circles will overlap in two places. You can either guess which square represents the sub – and risk losing a torpedo – or you can triangulate with a third circle, which should tell you which intersection hides a sub.
Just be careful that you don’t get confused when you’re actually pinging different ships! It’s better to do more pings to be certain rather than wasting one of your precious torpedoes.
Triangulation uses intersecting lines to reveal positions on a 2D plane. For example, imagine two mobile phone towers one kilometre apart. Each says your phone is one kilometre away. You can draw a circle representing a one kilometre radius from each tower, which will overlap in two places one kilometre away. To find which one represents you, you’d need a third tower nearby. If it says you’re 500 metres away, its own circle will overlap with just one intersection, revealing your location.

Enjoy this activity? Visit for more.

Brainteaser answer

Kelly’s next word is ‘half’. (Fourth is also another way of saying quarter!)
Written by Mike McRae ·  Edited by David Shaw ·  Formatted by David Shaw 
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