Did you solve it? Numbers in New Guinea | Mathematics


Earlier today I set you the following problem about how to count in Ngkolmpu, a language spoken by about 100 people in New Guinea.

Ngkolmpu does not have a base ten system like English does. In other words, it doesn’t count in tens, hundreds and thousands. Beyond its different base, however, it behaves very regularly.

Counting in Ngkolmpu

Here is a list of the first ten cube numbers (i.e. 13, 23, 33, …, 103):

1, 8, 27, 64, 125, 216, 343, 512, 729, 1000.

Here are the same ten numbers when expressed in Ngkolmpu, but listed in random order. Can you match the correct number to the correct expressions?

eser tarumpao yuow ptae eser traowo eser

eser traowo yuow


naempr ptae eser traowo eser

naempr tarumpao yuow ptae yuow traowo naempr

naempr traowo yempoka


yempoka tarumpao yempoka ptae naempr traowo yempoka

yuow ptae yempoka traowo tampui

yuow tarumpao yempoka ptae naempr traowo yuow


Ngkolmpu has a base 6 system. (Which is incredibly rare, and researchers believe may be a result of tallying yams).

When looking at all the numbers you may have noticed that when the expression has more than one word, the penultimate word is always traowo.

Also, whenever the expression has more than three words, the word ptae always appears two words before traowo. And whenever the expression has more than five words, the word tarumpao always appears two words before ptae.

This patterns leads us to think that the structure of writing a number is:

A tarumpao B ptae C traowo D

Since traowo is more common than ptae, which is more common than tarumpao, we are led to the supposition that traowo is the base, ptae is the (base)2 and tarumpao is the (base)3.

[The equivalent in English would be that our word expressions are ‘A thousand B hundred C tens D.]

With this hypothesis, we need to find the base. If the base is 7, then tarumpao is 343. But there are five expressions with tarumpao, but only four numbers 343 or above, which means that tarumpao cannot be 343. The base must be less than 7.

If the base is 5, then tarumpao is 125. There are five expressions with tarumpao, but six numbers 125 or above, which means that tarumpao cannot be 125. The base must be more than 5.

The base must be 6, with ptae = 36, and tarumpao= 216.

Thus naempr is 1, and so on by comparing digits we get the full answer: [the expressions are listed in the order 1000, 27, 1, 64, 343, 8, 216, 512, 125, 729.]

Another way you could have made a reasonable guesse that the base is 6 is to notice that there are 8 different Ngkolmpu words used. Since there are different words for the base, (base)2 and (base)3, the other five words are likely to be the other ‘single digits’: 0, 1, 2, 3, 4 and 5.

If you want to read more about Ngkolmpu, check out this blogpost by Matthew Carroll, which explains how yam tallying could have led to the base 6 system. You will also learn the unique words for 64, and 65, which are 1296 and 7776.

I hope you enjoyed the puzzles. I’ll be back in two weeks.

Thanks again to the UK Linguistics Olympiad, where this puzzle first appeared, and to its author Simi Hellsten. The UK Linguistics Olympiad is a national competition for schoolchildren that aims to encourage an interest in languages. It has different levels and is open to children of any age. If your local school does not already take part, you can join up through the UKLO website.

I set a puzzle here every two weeks on a Monday. I’m always on the look-out for great puzzles. If you would like to suggest one, email me.


I’m the author of several books of puzzles, most recently the Language Lover’s Puzzle Book. I also give school talks about maths and puzzles (restrictions allowing). If your school is interested please get in touch.

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