Story cube 3 - bumble bee, person dancing, a burst

On his way to collect nectar from the flowers, Bumble always liked to stop in the garden and watch the humans. They were strange creatures, with only two legs and two feelers, standing upright whenever they walked. The rest of the hive warned Bumble about the humans, but his curiosity always drowned out their voices in his mind. How bad could they be really? They couldn’t even fly.

There was the Man, the Woman and the Smaller One. Bumble didn’t know if it was male or female. There was nothing to set it apart either way. It spent all of its time making loud noises or sleeping.

Today, they were all outside. The Man was spraying water from a strange long, green thing all over the garden. The Woman was holding something in her feelers, staring at it non-stop, only pausing to flick something over then resume staring at it. Bumble had no idea what that was. The Smaller One was sitting in the grass, pulling it up and dropping it on its head. It made a loud gurgling sound each time that made Bumble feel happy. He didn’t know what the sound was, nor why it made him so happy, but he wanted to hear more.

He decided to get closer. Bumble usually kept his distance from the humans. He was unsure how they would react to him and obviously the warnings from the hive had had some effect on him. Today was just too much though. He had to know more.

As he got closer, he saw the Smaller One look up at him. It reached its pudgy feelers towards him. Its face didn’t seem evil. In fact, it looked like it was friendly, so Bumble landed on top of its feeler. There was no fur on the Smaller One’s feeler. It was just covered in a warm, fleshy substance that was kind of sticky. The Smaller One was bringing its feeler closer to its open mouth.

Suddenly, Bumble heard a high pitch noise come from the direction of the Woman. Then there was a waft of air that rushed past him, causing him to take to the air in shock. What was going on? The Woman was now holding the Smaller One, who was making that loud noise again. Its face was all screwed up and going red. Where was the Man?

Just as Bumble turned round to look for the Man, he felt another waft of air. This time he was pushed sideways. The Man was waving his feelers around. He looked ridiculous. Bumble had heard that humans sometimes moved like this when there were harmonious sounds playing, but he couldn’t hear any now. Bumble moved in and out of the Man’s flailing feelers, but eventually one hit him.

Bumble felt a smack and his vision flashed white before him. He felt himself soar through the air, but was powerless to stop it. He landed in a flowerbed with a bump. He lay there for a while, gathering up his strength, lying still in case the Man came looking for him. He didn’t want to be found.

He had no idea how long he lay there, but after a while he could no longer hear the humans. They must have left the garden, so Bumble got up slowly and flew to the top of one of the flowers. After a quick glance around to make sure he was really safe, Bumble shot off back in the direction of the hive.

He would never stop to watch the humans again.

Weekly Update

So far this week, I have done:

1 - Write story cube story 5 (bonus: try and write the one I missed too)
2 - Write types of numbers post
3 - Find more ideas for future types of numbers posts
4 - Go to work experience on Monday with enthusiasm
5 - Come out of work experience on Monday feeling like I have achieved something
6 - Go for a run - hopefully without too much knee pain
7 - Go on an adventure to find an archery store
8 - Read a chapter of a book - any book, I don't read enough
9 - Go through tax returns with the boyfriend's mum
10 - Do my archery exercises everyday

Well, the story cube is just a draft, but I've written it a lot earlier than I usually do, which is a big improvement on my failure to do it last week. I managed to read a chapter of my Bedside Algebra book and we found the archery store that had been recommended to us. Mum bought me an arm guard and a finger tab as the birthday present she owed me. We didn't get any more than that because the man in the store happens to be coming to the club that we shoot at, so he said he would be able to help me out more after he has seen me shoot. Who knows, maybe this time in a couple of weeks I will have my own bow! I'm not going to lie, that would be awesome.

I have done my archery exercises every day so far, except today, but that's because I did actual archery so I think I can forgive myself for that one. I could definitely tell that the exercises I have been doing are making a difference to my strength. I even almost have a muscle on my right arm!

Tomorrow I start my work experience. I'm quite nervous about it because my brain likes to go into overdrive and over think everything. If I don't know exactly what is supposed to happen, I start to panic. I don't know if I am driving, or if I can get a lift (which I would prefer - one less thing for me to worry about doing). I don't know where to park if I am driving. I asked, but I'm not sure I know which car park they meant. I don't know what it is going to be like working there. I think it is going to be interesting, but until I've actually done it, I am in try-not-to-panic mode. I'm going to write a post about it once I've done a week or so of work, so hopefully then I'll be able to say there was really nothing to panic about!

Story Cube 2 - person playing with dolls, walking stick, person jumping off a ledge

Christie sat on the floor of her grandparent’s living room, playing with some dolls that had been left out for her. She didn’t really like dolls, but they were the only toys left over from when her mother was young, so they were brought out every time she came to visit.

What Christie really liked was stories.

She got that from her grandfather, who was currently sat in his usual spot in the big armchair, in front of the fire, reading a newspaper. Propped up against the wall beside him stood his gnarly old cane. Christie stared at it a while, then turned to her grandfather.

“Grandpa, why do you walk with a cane?” she asked him. He lowered his newspaper slightly, and looked over his glasses at her, with a hint of a smile on his face and a gleam in his eye. Christie knew the look well; it was story time.

“Well, you see sweetheart, I wasn’t always the doddery old man you know me to be. I was once an adventurer. I travelled all over the world looking for new and exciting journeys to take that would lead me where I least expected them to.”

Christie listened intently. Grandpa always told the best stories, and she could tell that today was going to be no exception.

“I was a young man, sailing across the sea as part of the crew on a small ship transporting a grand jewel back to England – the Wimueko. This jewel was priceless, so of course we had to guard it with our lives.
“All was going well until one fateful day. We had been suffering from a lack of wind, so had been rowing all morning. Rowing a ship, even a small one, is very hard work, so as you can imagine we had to stop for a rest. It was at this moment, when our guard was down, that we were suddenly ambushed by pirates! We rushed to arm ourselves, but the pirates were too quick. They had us surrounded. We were only a small crew, so they tied us to the mast of the ship. Then the captain arrived.

“Captain Ferdinand was the most feared pirate in them days. I recognised him at once by the scar across his nose, and the fact that he had different coloured eyes. One was a watery blue, but the other was a deep, dark black. There was evil in that eye.”

Christie shuffled closer to her grandfather’s feet. “What did you do Grandpa?”

“I did what any brave adventurer would do with something under his protection.” He leaned over his knees, close to Christie’s face, and whispered dramatically, “I challenged him to a duel!”

Leaning back in his chair so that he could gesture wildly with his arms, he continued with his story.

“There was just the two of us, circling each other, cutlasses in hand. All the other pirates gathered round to cheer on their captain, waiting for bloodshed. My blood. The fight began and we slashed at each other, high and low, jabbing and slicing, parrying and blocking. Just as one of us got the upper hand, the other would push back and level the playing field. It was a tremendous battle.

“Eventually, I got Captain Ferdinand cornered and pinned to the floor. The battle was won, and I was the champion. I stood over him and told him to leave and never come back! The pirates began leaving our ship, the captain following last. We thought we were safe, but at the last moment, the dirty rotten pirate turned and sliced at my leg, then left laughing on his ship.

The cut went deep, and became infected on the way home, but luckily the wind was on our side and we made a swift journey back. My leg was treated, but I have walked with a limp ever since.”

At this point, Christie’s grandmother poked her head through the door, announcing that dinner was ready. Christie jumped up and grabbed the cane for her grandfather then watched as he lumbered his way into the kitchen. As much as she loved hearing her grandfather’s stories, Christie also had a thirst to know the truth. Walking over to her grandmother, she whispered to her so that her grandfather didn’t hear, “what really happened to his leg Grandma?”

Christie’s grandmother looked over at her husband, then back to her granddaughter.

“Well,” she whispered back. “When he was a boy, he thought he could fly, so he tried to jump off the roof.”

Weekly Targets 6th August - 12th August

After the horror of realising that Sharknado 2 was airing the same time as in America, not when I thought it was, and watching the Shining instead, I accidentally forgot to set myself targets last week. I did however start a new thing with Liberty. She sent me a link to a post in relation to helping my time slippage, which was about this schedule that the person who wrote it sticks to to stay productive. We thought we would try it out.

I let Liberty fill it in today... Basically it is just a laminated schedule that I can write on each day with whiteboard pens. It is quite hard to stick to it exactly at the moment because each day I'm doing something different, but I think it will be really useful when I go back to Uni.

Anyway, this week I managed to go driving a couple of times (scary, I know!) even once on my own today (even more scary!!). I had an awesome time shooting balloon dogs at archery, with the instructors saying that I had advanced technique and joking that I could be the next Danielle Brown. I went to Ford Hall Farm and have sorted out getting some work experience there starting next Monday. I finished the second chapter of the accountancy book. Now onto this week:

1 - Write story cube story 5 (bonus: try and write the one I missed too)
2 - Write types of numbers post
3 - Find more ideas for future types of numbers posts
4 - Go to work experience on Monday with enthusiasm
5 - Come out of work experience on Monday feeling like I have achieved something
6 - Go for a run - hopefully without too much knee pain
7 - Go on an adventure to find an archery store
8 - Read a chapter of a book - any book, I don't read enough
9 - Go through tax returns with the boyfriend's mum
10 - Do my archery exercises everyday

Can you tell I'm getting slightly obsessed with archery?

Types of Numbers - Taxicab numbers

There was a mathematician named G. H. Hardy, who discovered and then later worked closely with a brilliant young mathematician named Srinivasa Ramanujan. One day in 1918 or 1919, Hardy went to visit Ramanujan in his private nursing home. Hardy said about Ramanujan:

"I remember once going to see him when he was lying ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me a rather dull one, and that I hoped it was not an unfavourable omen. 'No,' he replied, 'it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways'."

 This can be generalised to the following definition:

Ta(n) is the smallest number that can be expressed as the sum of 2 positive cubes in n distinct ways.

These numbers have been dubbed Taxicab numbers in honour of this anecdote. At the moment, only 6 taxicab numbers are known. The first few, with the cubes that make them up, are:

Ta(1) = 2 = 1^3 + 1^3

Ta(2) = 1,729 = 1^3 + 12^3
                      = 9^3 + 10^3

Ta(3) = 87,539,319 = 167^3 + 436^3
                                = 228^3 + 423^3
                                = 255^3 + 414^3

The remaining three known taxicab numbers are 6,963,472,309,248, 48,988,659,276,962,496 and 24,153,319,581,254,312,065,344.

I would love to know how they discover which number is the smallest one. Is is just a brute force algorithm that goes through every possible combination of cubes for each number? Or are there ways to narrow down the variables? At a guess, I would say it is mostly brute force, as there has been a lot of time between the discovery of each new number, the latest one only being discovered in 2003.

Although we only know the smallest number for six equal sums of two cubes, it has been proven by Hardy and Wright that the number of sums can be made arbitrarily large. In other words, the list of taxicab numbers goes on forever. There is not a biggest taxicab number.

Some people take taxicab numbers to be the list of numbers that are the sum of two cubes in two or more distinct ways. The first few are 1729, 4104, 13832, 20683, 32832, 39312, 40033, 46683, 64232, ...

I can see how you could find these numbers. You would have to limit the number you look up to, say to N. Then you take every cube number between 1 and N and add them to 1. Next add every cube number between 2^3 = 8 and N and add it to 8. Do the same up until N^3 + N^3, then sort all of the sums into order of the answers. Then you can go through the list and see which sums have the same answers. Obviously this would take very long to do by hand, but hey, that's what computers and algorithms are for!