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. 1^{3}, 2^{3}, 3^{3}, …, 10^{3}):

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*

*naempr ptae eser traowo eser*

*naempr tarumpao yuow ptae yuow traowo naempr*

*naempr traowo yempoka*

*tarumpao*

*yempoka tarumpao yempoka ptae naempr traowo yempoka*

*yuow ptae yempoka traowo tampui*

*yuow tarumpao yempoka ptae naempr traowo yuow*

**Solution:**

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 6^{4}, and 6^{5}, 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.*