This is my third and final blog post about my trip to Barbados over Christmas and New Year with my husband, Howard. Apart from enjoying the scenery, meeting the people and eating lots of delicious food I also came across some interesting numbers. One of them was on the little minibus that you can see above. These are private and have routes all over the island and a fixed fee of 350 Barbadian (or Bajan) cents no matter how far or where you’re going. 350 Bajan cents is about 175 US cents and about £1.45 Sterling. Our most regular trip was a couple of miles down the highway (only a single-lane road) to the next town – Oistins – for shopping or for the famous fish fry. We had our tandem bike with us, but this particular road wasn’t safe as these little buses go very fast and often pass without much room. The buses were incredibly flexible. We stayed in a Bajan neighbourhood where there weren’t many white people: the likelihood was that if we were on foot we’d be looking for transport. If we were walking up one of the side-roads to the highway and one of the minibus drivers spotted us they would beep their horn and we would wave if we wanted a ride. They would then, literally, reverse down the road towards us so that we could hop on. Clearly, because these buses are privately owned, the drivers have an incentive to gather as many customers as possible. Another brilliant thing was that they were very frequent – every three to five minutes at the most. Also, when travelling back to our lodgings, it was sometimes the case that we’d be dropped right outside our door! We were told that many of the drivers knew our neighbour, Juliet, and that if we let them know that she lived upstairs from us they’d be sure to know where to drop us. This was wonderful. One day we travelled from where we were staying to the capital, Bridgetown, ten miles away – all for 350 Bajan cents each! I keep telling people about these minibuses in the hope that something similar can be dreamt up in the UK.

The little buses weren’t particularly safe – no seatbelts and often not much to hold on to. On one it was so crowded that I was worried that Howard would fall out of the door as it wasn’t always closed at the point of setting off. After this we always waited for a bus that wasn’t crowded – there were so many that we knew we wouldn’t be waiting long. Bajan people are really friendly, and we often got chatting during our journey.

For this blog post my number will be 350. If this is the first of my posts you have read then the idea is to identify a number that occurs naturally in my everyday life and then find an interesting mathematical property to share. 350 is a Zumkeller number – I’d never heard of this before – and there are details below.

Zumkeller Numbers

A Zumkeller number is one whose divisors can be partitioned into two sets with the same sum.

For example, 6 is a Zumkeller number. Its divisors are 1, 2, 3 and 6. These can be partitioned into two sets with the same sum as follows:

1. 1, 2, 3 where 1 + 2 + 3 = 6
2. 6

Another example is 20. Its divisors are 1, 2, 4, 5, 10 and 20. These can be partitioned into two groups as follows:

1. 2, 4, 5, 10 where 2 + 4 + 5 + 10 = 21
2. 20, 1 where 1 + 20 = 21

Looking at 350: its divisors are 1, 2, 5, 7, 10, 14, 25, 35, 50, 70, 175, 350. These can be partitioned into two sets with equal sum as follows:

1. 5, 7, 10, 350 where 5 + 7 + 10 + 175 = 372
2. 1, 2, 14, 25, 35, 50, 70, 175 where 1 + 2 + 14 + 25 + 35 + 50 + 70 + 175 = 372

The first Zumkeller numbers are 6, 12, 20, 24, 28, 30, 40, 42, 48, 54, 56, 60, 66, 70, 78, 80, 84, 88 and so on.

All Zumkeller numbers are perfect where a perfect number is a positive integer that is equal to the sum of its proper divisors (i.e. excluding the number itself). This means that the divisors of a Zumkeller number can be partitioned into one containing the number itself and the other containing all the other divisors – such as in the case of 6 above. The reverse is also true: all perfect numbers are Zumkeller numbers.

If a Zumkeller number is not perfect, then it’s abundant: an abundant number is one where the sum of the proper divisors is greater than the number itself.

12 is a Zumkeller number and is abundant because its proper divisors are 1, 2, 3, 4, 6 and their sum is 1 + 2 + 3 + 4 + 6 = 16. These can be divided into two groups with equal sum as follows:

1. 1, 3, 4, 6 = 1 + 3 + 4 + 6 = 14
2. 2, 12 = 2 + 12 = 14

Unlike perfect numbers not all abundant numbers are Zumkeller numbers. For example, 72 is abundant but not Zumkeller. The divisors of 72 are 1, 2, 3, 4,6,8,9,12,18,24,36 and 1 + 2 + 3 + 4 + 6 + 8 + 9 + 12 + 18 + 24 + 36 + 72 = 195 which cannot be partitioned into two. In fact, as far as I can tell, all abundant numbers that are not Zumkeller have odd proper divisor sums. Do you have any thoughts about Zumkeller numbers? Please share below.

Also, the word Zumkeller means ‘into the cellar’. Could it be applied here because it is easy to partition a Zumkeller number and store it in a cellar? I’ve no idea! How about you?