A few months ago, I published a maths game for year 3-4 kids (called Cross The River).
This was a fun enough game, but was aimed at kids who are already pretty confident with multiplication. But I wanted a game for kids who are at the beginning of their multiplication journey. I wanted the game to achieve the following:
Excite them, so that they wouldn’t realise that they were just doing rote learning.
Expose them to a full range of multiplication problems, from 1 to 10 times tables.
Encourage them to say the problem and answer out loud, to aid memorisation.
Allow them to play, even if they haven’t learnt all of their times tables well (i.e. have the answer available, so they can still play the game).
So I created the following two-player game (see also PDF here: Capture The Flag , sized for single A4 page print). This game is suitable for children in year 2-3, who are still working on some of their times tables, as well as older kids who know their times tables but will become faster if they can do some extra repetition.
I haven’t run the numbers to see how many turns it takes on average to complete the game, but my gut feeling is that it will be on the order of 15-30 turns… I’ll come back later and calculate the statistics at a later date.
I hope you enjoy!
Feel free to share with that recalcitrant primary-school child in your life, who needs to be encouraged to do more maths.
(P.S: If you laminate the sheet, you can use whiteboard markers on it, and re-use it again and again)
(P.P.S: While I included in the game sheet a link to an online dice roller, to roll two 10-sided dice, I definitely encourage you to go to your local game shop and ask to buy “2d10″… They’ll know what you mean. In fact, while you’re there, you might as well pick up a few d20, maybe 1d12, 2d8 and 3d4, just for fun)
If you can remember all the way back to mid-primary school, you might recall that learning multiplication required a lot of rote learning, and wasn’t ALWAYS the most exciting activity. I don’t personally recall any such suffering, but my relationship with numbers is different to most people’s. According to my son, however, there’s a fair bit of repetition.
There are plenty of board games that involve addition and subtraction, but relatively fewer where each turn requires the multiplication of two random numbers together (thus infusing the game with sneaky maths homework)… So I created the following game (see also PDF here: Fun With Numbers – FINAL v3.0 , sized for single A4 page print), which turns that multiplication into a board game.
The outcomes of playing a single game are:
Each turn requires the child to multiply two random numbers (between 1 and 10) together.
Average of 21 turns per game (if you limit each game to a maximum of 30 turns)
20% of games will be “easy” (less than 15 turns)
20% of games will be “you lose!” (more than 30 turns) – I felt that putting a turn limit on the game added stakes (it’s possible to lose the game), and limits how long the kid has to play (so that they don’t get bored).
The above statistics, and the design of the game board itself, was the result of me going way further down the rabbit-hole than required for a simple homework activity, and involved excel spreadsheets, VBA coding, wild guesses, stochastic simulations, genetic algorithms, drawing illustrations, and good clean fun with numbers.
I hope you enjoy. Feel free to share with that recalcitrant primary-school child in your life, who needs to be encouraged to do more maths.
(P.S: While I included in the game sheet a link to an online dice roller, to roll two 10-sided dice, I definitely encourage you to go to your local game shop and ask to buy “2d10″… They’ll know what you mean. In fact, while you’re there, you might as well pick up a few d20, maybe 1d12, 2d8 and 3d4, just for fun)
(P.P.S: Theoretically I can use the same framework to do the same game for multiplications up to 12×12, or any other number, and I do also have harder versions [which take more turns to complete], which I could publish… but I’d probably only do that if there was a teacher or parent in need. Drop a comment if you are a teacher or parent and you are in need)
(P.P.P.S: I realised that teachers don’t want to waste a bunch of ink, so I have also created a copy of the sheet with less shading in the cells, Fun With Numbers – FINAL v3.0 – Low Ink Version, which can be printed in black and white to save cost. Although it might be better to just print a couple and laminate them for use with whiteboard markers)
If you like this graph and associated commentary, remember to check out the rest of my graphs on this blog, here: Graphs of the Day
This post is about the most beautiful spreadsheet I’ve ever made… Possibly the most beautiful spreadsheet ANYONE has ever made. Skip to the end for pretty pictures…
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So, I am going to assume that you’re all familiar with the term “Nerd Sniping”. Yes? Good.
Well, those who know me well, know that I’m fairly susceptible to being nerd sniped, and I have a habit of sniping myself. I’ll have some idea and won’t be able to fully shake it until I’ve gone down the rabbit hole. And after my last post, regarding hexagonal grids, I went down a deep rabbit hole. And I mean DEEP.
You see, it started with the innocent thought “I could probably write some code in an excel spreadsheet which could draw these for me…” for you see, when all you have is a hammer, everything looks like a nail. Excel is my hammer. Any I was right: I COULD write such an excel spreadsheet. SHOULD I though? Probably not. I had no such need. But when some people say “why” I say “why not”. And then when I’m given a long list of reasons, I’m like “Bah! stop being a spoil-sport!”
I don’t want this post to be too long, so I’ll give a quick bullet point list of my thought process and steps taken, then show some of the images that my spreadsheet spat out, including some more comparison to hexagonal grids, square grids where diagonals count as 1 movement, and square grids where every second diagonal counts as 2 movements.
“I can write some code to fill out a grid with the number of moves from one or more ‘goal’ cells. That’s just a progressive wavefront algorithm.”
[writes some VERY sloppy, inefficient, but quick-to-produce code to do so for the simple case of each diagonal equaling one movement]
“Hmmm. Should be easy to turn that into code where the diagonals are worth 1.5 movements, then round them down afterwards to match the ‘optional rule’ discussed previously”
[Ended up scrapping the original code because it was too sloppy, and didn’t work for cases where diagonals were treated differently to up\down\left\right movements. Get frustrated by bugs. give up and decide to work on the hexagonal grid]
“Surely there must be people online who have written some code that I can steal, for creating hexagonal grids in excel”
“What? Why are these idiots talking about needing three-dimensional coordinates for a hexagonal grid? Do they not understand that it’s still just a 2 dimensional grid, and therefore only requires two parameters to define any position on the grid?! change the coordinates from DOWN and RIGHT, to DOWN and DIAGONAL. EASY! WHY ARE YOU ALL FOOLS! JUST GIVE ME CODE I CAN STEAL!!! ARGH! … fine… Who needs you anyway, Stack Exchange”
[write the code to draw a hexagonal based grid, centred on coordinate 0,0, using Shapes in excel]
[Add code for automatically colouring cells]
[write much neater code for checking all unvisited cells, and incrementing their distance from the Goal cells]
[write code for copying the old Goal and Barrier cells from an existing grid, and pasting them into the new grid, when resizing]
“Ugh, it’s too much of a pain typing ‘B’ into these cells when I want to draw lines of barrier cells”
[have a win, and write some simple code to enable me to click a cell to turn it on\off as a barrier cell]
“Wow!… That is a very, very pretty spreadsheet”
[Spend literal hours marvelling at how pretty my spreadsheet it]
[Tell anyone who will listen about how pleasing hexagons are to the eye]
“Wow! I’m Done! I can share the images with the world!… Crap, I forgot about the other square grid spreadsheet”
[Hours more coding, which would have been faster if I hadn’t kept forgetting that VBA in Excel reverses the coordinates for cells, going row\column, instead of column\row]
So, I’ve done a couple of side-by-side comparisons between the Hexagonal grid and the two square grids (with\without discounted diagonal movements), which shows the same basic outcomes as last time: counting diagonals as 1 movement significantly increases how far a diagonal move will get you; hex and the “optional diagonals” rule (from D&D) are fairly similar.
Basic rule (diagonals worth one movement)
Optional rule (every second diagonal worth 2 movement)
Hexagonal grid, all movements worth 1 movement
Rules as Written
Optional Diagonals
Hex
Once I’d done this comparison, I then circled back to some thoughts I’d had about about how hex grid D&D maps needn’t be clunky and have weird edges: That’s an artifact from us trying to apply our “buildings should be rectangles” mentality onto a fantasy world. Why not have triangle and hexagon based structures? So I started playing around with building some crude maps using my hex grid map generator (nerd sniped myself again: now I’m going to add additional features to the “on click” feature, so that I can add different types of tiles to the grid, and get it to automatically apply colouring or even images, turning it into a map creation tool)… Here are the results. I didn’t go fully down the rabbit hole with thinking about what a hexagonal shaped building or cave system would REALLY look like, I just had fun. I did find that when I was being more freeform, the hexagonal grid gave nice cave-like tunnel-ish shapes.
And here are some pictures that my daughter and I did just for fun, while I was playing with the wavefront algorithm. My daughter says that they look like snowflakes.
My son says that they look like stars, or like I’ve been making spreadsheets to show how fast rocks are moving… Because I recently showed him some of my Discrete Element Modelling simulations from work, in which the rocks are coloured according to their speed, in very similar colour scales…
If you like this graph and associated commentary, remember to check out the rest of my graphs on this blog, here: Graphs of the Day
This is a long one! Skip to the bottom for graphs, links to the 3D model, and links to useful graph paper printing tools.
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So, before I start, I’d like us all to just take a moment. Take a moment, and consider how pleasing this 3D printed item is:
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The pleasing hexagons, the isometric cubes that aren’t there, the patterns within patterns within patterns…
Sigh…
…
Okay, this post isn’t just about how nice geometry is (very), but this particular item was designed and printed for a very practical reason, and it struck me, when I picked it up and held it for the first time, that it’s also very pleasing to the eye. Not beautiful as such, just… Pleasing.
The link to the 3D model for this hexagon template is at the bottom of this page, after a long and rambling explanation as to why I printed it and why I needed hexagons to begin with.
Okay, so what is this post actually about? First of all, don’t get put off you’re not into Dungeons and Dragons, which this post is superficially about. It’s more about geometry and pretty coloured graphs, and also a little bit about how I don’t understand why sometimes people don’t understand things that seem to make sense… but it’s also about overcoming our currently held beliefs… or at least interrogating them. Just scroll down to the graphs, if you want.
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Since I started playing D&D again, I’ve seen many people lamenting (on the internet) about things which don’t really seem worth lamentation. I won’t go into detail on all of them (many of them seem to stem from people’s inabilities to hold onto medium sized groups of numbers), but one that stood out to me was “issues” when measuring a character’s movement when going diagonally… I fully went down the rabbit hole on this one.
So for those who don’t know, when you’re using a grid and miniatures in D&D, typically for combat\exploring a dungeon, your characters are allowed to move a certain distance each turn. That distance is typically measured in increments of 5 feet (and it hurts me every time I have to measure in feet, but that’s a rant for another day).
Now, one option is to use no grid. You can measure your movement using a ruler of piece of string. This gives you the greatest amount of freedom and flexibility, but also makes measuring distances more difficult than any of the other options, because you can’t just count the grid spaces. I will ignore this option for now, partly because I don’t enjoy measuring out distances on the map, partly because having a grid gives you an immediate and consistent sense of scaly, but mostly because grids are pleasing and bring order to chaos.
So, assuming that you’re not some gridless animal, the most common grid style is a (typically 25mm) square grid, with each 25mm square being equivalent to 1.25m (5 feet, if you really insist). So counting movement is easy, because you just step your miniature from square to square, counting the distance in *shudder* 5 ft increments… But what about diagonals? well, each diagonal movement is worth 7.072ft, right? Well, that introduces a level of accounting that most players don’t like; we don’t ACTUALLY like thinking of our characters as being able to move (typically) 30 ft, we like being able to think of it as moving 6 squares!
So the basic rules of D&D (5th edition) is that when you move diagonally, it counts as one square of movement (i.e. 5 ft). That’s it. That means that when you move diagonally, you get a bonus 41% distance travelled. Sweet!
The optional rule (in the Dungeon Master’s Guide) is that every second diagonal movement counts as two squares (i.e. 5 ft, 10 ft, 5 ft, 10 ft). So if you move two squares diagonally, you lose 6%. Meh, close enough.
The other option, is to use a hexagonal grid, where you only have 6 neighbouring squares, all of which are 5 ft apart (centre-to-centre).
So which method should you use? Well, There’s no “correct” answer, and I’m not even sure if it would be helpful to do a pros and cons list, or bullet points for each method. Instead, what I’ll do is present my graphs, then step through my train of thought, and maybe by the end I will have arrived and a position on this topic.
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The following graphs shown three methods for movement, side by side:
Rules as written – diagonals count as 1 square of movement
Optional rule – every second diagonal counts as 2 squares of movement.
Hexagonal Grid – every movement can only be 1 square away.
I’ve colour-coded each square based on the minimum number of “steps” from the centre point. Ooooh. Pretty…
Rules as written – diagonals count for 1 square of movement
Optional rule – every second diagonal movement is worth 2 squares of movement
Hexagonal grid
A few initial observations:
This scheme, calculating the shortest distance from one location to another by incrementing the distance based on the shortest of the surrounding cells, is (I believe) called a “progressive wavefront” algorithm. Fun stuff, worth a read if you ever have need to program a robot to follow the shortest route from one location to another, and has the ability to be modified to push\pull the route based on areas that are preferred or non-preferred for travel.
The hex grid is BY FAR the most beautiful
The standard (rules as written) method gives a significant increase in “actual versus measured” distance. If you’re trying to outrun a circular area of effect spell (i.e. fireball): run diagonally!
While not quite equal, the optional rules give a similar result to the Hex grid, and are far closer to being circular (as expected).
None of these reflect the fact that you would, in real life, tend to run in all sorts of diagonals, not just 45°… but the Optional Rule method gets you pretty darn close to a neat circle.
But I felt that that didn’t tell the whole story. What if there’s an obstruction? a table, or a stalagmite, or a pile of unspeakable filth left by goblins (or elves, #everybodypoops) with poor sanitation practices?
So I did another version of these graphs, showing the effect of an obstruction. I kept the obstructions in a similar shape\location for each graph, but it was late and I made some copy-paste errors, so it’s not perfect. I’ve accepted this and I’m not fixing it; either cut me some slack, or take your complaining elsewhere.
Rules as written – showing effect of obstruction on movement distances
Optional rule – showing effect of obstruction on movement distance
Hexagonal grid – showing effect of obstruction on movement distance
The obstructions further exacerbated the effect, exactly as you’d expect: The more zigs you zag, the more the error effects add up.
So, what are my thoughts on the CORRECT grid style and method of counting diagonal movements?
The Optional Rule is the most accurate, but I understand that some people don’t want to\can’t get their head around the extra counting.
The Hexagonal grid is more accurate than the square grid, without adding extra maths.
The square grid is better suited for maps of buildings, where the grid will line up with the structure… IF your structures are typical rectangular structures.
There isn’t any reason that your dungeons can’t be triangular or hexagonal based, with hallways aligning with the directions of the hexagon. This could in fact make your maps more interesting and less ordinary. Consider it.
For natural features\terrain, nothing lines up with a grid anyway, so Hex is as good as square.
Square grids are easier to draw. Hex grids are a pain in the butt, requiring more marking out and measurements, unless you have some kind of template to help you draw them… ENTER MY TEMPLATE! See below.
If the same rules apply to all characters and movements, then it doesn’t actually MATTER that your diagonal movements are no longer perfectly correct. It’s all slightly abstract anyway, so maybe our square grid doesn’t map perfectly onto a square cartesian surface, maybe it’s actually a curved representation of the “actual” areas being depicted. I bet there’s a really interesting maths paper in this (if anyone has ever read such a paper, please get in the comments). What I’m saying is that if a given path between two points is the same measured distance for all entities, then that is just the distance that it is between those two points! The space is curved in such a way that this is the internally consistent coordinate system for that world, you’re just seeing it wrong on your cardboard grid. As long as everyone uses the same rules, and thinks about their options in line with those understood rules, then… it doesn’t actually matter.
So, when I picked up one of those cheap jigsaw puzzle mats (the cheapest of green felt-like material) from the shops, and naturally decided that I could give it a second purpose as a sheet of grassland terrain for playing DnD-ish games with the kids, I had to go through this internal struggle. Eventually, I decided that:
I think I will generally stick with square grids for buildings\structures, because I’m used to dealing with rectangular structures
When drawings maps on paper\computer, I will consider hex grids from time to time, to make the structures more interesting (triangular or hexagonal shaped)… but only if I have hex grid paper or software available.
For natural terrains, I will use hexagonal grids, but only if I have a good template for drawing the grid.
So before I started drawing the grid on my new green jigsaw puzzle mat, I designed and 3D printed a template that allows me to easily align and mark out the corners of the hexagons (which I may leave as is, or may join up to form solid edges). I designed it to give hexagons which are 25mm across the flats, with holes that are ø3mm. I should have gone 4mm, as it’s just a touch to snug for the permanent marker that I wanted to use, so I’ve uploaded versions with 2mm, 3mm and 4mm holes\slots. Enjoy!
Here’s a handy tool that I use, to print paper grids (either square or hex) of whatever size you want:
If you like this graph and associated commentary, remember to check out the rest of my graphs on this blog, here: Graphs of the Day
So apparently I had previously uploaded an old version of this UPDATED version of my personal space diagram. I had not removed the “handshake” zone from the image. I have now corrected this terrible mistake, showing the correct “foot five zone”.
Now, quick point of clarification for people who have adopted an “elbow bump” instead of a handshake, during the covid-19 era: Stop it. You’re wrong.
I understand the motiviation here, with people trying to not touch each others hands, since hands are where we carry out germs (or lots of them, anyway). We use hands to touch the world, touch our food (depending on the food, or the quality of our upbringing), we touch everything! So not also putting our hands on each other is a good idea. However… by doing an elbow bump, you have to bring your face closer to the other person, which makes it easier you to shoot a droplet of saliva and germs onto the other person. Also, we’re constantly being told to cough into our elbows, so that we’re not coughing on our hands (spreading germs to everything we touch). So by doing an elbow bump, you’re saying “hey, lets get the bits that we cough on, and rub them together!” …and that’s just weird.
Also, does anyone else have an issue with being told to cough\sneeze into their elbows to begin with? I mean, yeah, it means you’re not sneezing on your hands, but… I just find it so much harder to hold a tissue with my elbows…
Mirth.
Anyway, the point is this: Switch to a foot five. I’ve been using the foot-five for a long time to show affection to people whom I don’t want to hug, and I think you all should too. If it’s not obvious to you, here are the reasons why:
Captain Simian And The Space Monkeys (mid-90’s cartoon. If you aren’t familiar, you have my condolances)
When you stick your foot out to foot-five, you naturally lean back, away from the other person. Yay social-distancing!
Shoes are already considered “kinda dirty”, so no-one is handling the soles of their shoes without washing their hands afterwards (or if they do, they ought to know better).
You aren’t sneezing into your shoes or handling them a heap. Sure, they have plenty of every-day germs and bacteria and dirt, but they aren’t where I’d expect to find covid hiding out.
If you like this graph and associated commentary, remember to check out the rest of my graphs on this blog, here: Graphs of the Day
Finally, a socially acceptable excuse to be more anti-social!
Okay, so this is just an update to my previous graph on personal space. ALL of the old lines are no longer applicable, just pay attention to the RED line.
I don’t know why people at the shops are so bad at social distancing. It’s not that hard: Son’t go close to other people, and be considerate by helping people not get too close to you! i.e.
Are you close enough to shake hands with someone? Then you’re too close!
Stick close to the wall, rather than stopping in the middle of a walkway at the shops.
If you’re in a shop with narrow aisles, don’t just go squeezing past people to get through, wait until they move on.
… Look, I’m not going to give you rules for every situation, just remember to stay 1.5m from other people.
“The process that requires the fewest things to be done correctly is the most likely outcome.”
(An adaptation of Occam’s Razor)
This means that when trying to anticipate a series of actions or events, where each action can be done correctly or incorrectly (or not at all), the most likely outcome is that which requires people to choose do to the right thing the fewest times.
This differs from Occam’s Razor in that the number of assumed events\entities may be the same (i.e. not multiplied beyond necessity), but the Saggers-Aitken Razor helps to further narrow down these otherwise equal options.
The Saggers-Aitken Razor is intended for prediction, and is great when thinking about Safety In Design, but may also be used for retrospective analysis in some cases.
An example of this is a discussion I had recently about Lego. Or more accurately: “Lego”, or “building-blocks”. You see, at Christmas I bought a large Lego-Compatible building block set from overseas for my little family to build. The kit was for all intents and purposes identical to the same Lego set, however without “LEGO” stamped on the tops of the blocks. I was telling a coworker about the quality of the kit, and my general review was “the blocks had great tolerances, so the fit together just right, although they were maybe not QUITE as well fitting as genuine Lego blocks… although that could have just been my imagination, because I was expecting them to not be as high quality as Lego”. You see, Lego need to have very high tolerances on their injection moulding processes, because if the fit (between the bumps and the holes) is too tight, the blocks are too hard to separate; too loose, and the blocks fall apart on their own.
I told my colleague that I suspected that when the moulds had worn out (as they slowly do, through normal use), they were discarded by Lego… and then probably taken right next door to the next injection moulding company, by some less-than-scrupulous worker, and used to manufacture reasonably good immitation blocks… Of course, I have no evidence to confirm or even suggest this, but it makes sense as a thing-that-would-happen. He said that Occan’s Razor suggests that it’s probably correct. Although, he would expect that Mr. Lego would have policies in place that the moulds should be destroyed, rather than discarded, to prevent exactly that from happening.
“Hmmm” I said “Not Occam’s Razor… I’d say ‘whatever relies on people doing the right thing the least number of times is probably the truth'”.
And thus the Saggers-Aitken Razor was born
The Saggers-Aitken Razor:
“The process that requires the fewest things to be done correctly is the most likely outcome.”
There is no system of storage that doesn’t start with figuring out what it is that you’re trying to store. Anything else is just “cramming stuff in a cupboard and hoping that it doesn’t explode in an avalanche of unknown matter which ends in a newspaper article titled “hoarder dies, buried by tonne of half empty tomato sauce bottles”, which no-one will read because: who buys a newspaper any more?
Here’s how organisation works:
Identify a need to store something
Figure out exactly what the something is
Figure out how much of that something you need to store
Locate potential storage locations for some something, and measure storage capacity of that somewhere.
If you have more something than you have somewhere, either find more somewhere, or reduce the amount of something
Put the something in the somewhere
Make sure you’re able to get the something out of the somewhere when you need it, ideally not leaving the somewhere as a mess of something in the process.
and that’s basically it. Nothing controversial there, and yet apparently I’m the insane one.
So, what’s this blog post about anyway? Was I talking about anything or going anywhere before I started ranting about somethings and somewheres? Yes, actually, I was going to talk about my shirts. They are the something, and the somewhere is my wardrobe.
For whatever reason, I decided that yesterday was the day to finally deal with the mess that was my wardrobe. So, naturally, the first thing I did was draw up a list of all the clothing items I need, and how many of them I need. This is the part which, according to my wife, is a little bit odd.
Here’s a table of the quantities that I finally settled on:
Jumpers
Decent – 2
Vest\Hoodie – 2
Exercise – 1
Jacket – 1-2
Shirts
Weddings, etc – 3
Work – 7
Casual, no buttons – 2
Polos
Decent – 2
Work – 1
Tee-Shirts
Casual – 6
Exercise – 3
Short sleeve button shirts – 2
Singlets – 5
Undies – 9
Boxers – 3-4
Pants
Jeans – 1-2
Work – 3-4
Casual – 2-3
Shorts
Casual – 4
Exercise – 3
Swimming – 2
Trackies – 2-3
Socks – 9
Site clothes – 2 sets
Cold weather (snow) – set (incl beanie)
There are based on the typical usage frequency, and the minimum wash cycle frequency that I’d be happy with. These numbers are significantly lower than the numbers that I USED to have (before the big clean-out), but all it means is that previously I could get away with washing the clothes less frequently, or more likely washing with the same frequency, but building up a large pile of clean clothes without actually putting them away.
So, now that I’ve identifies that I can get rid of a heap of clothes, how do I decide which ones no longer have a place in my life? Very simple, I use the following hierarchy to help rank the clothing in each category, then get rid of the lowest scoring items in excess of the required quantity:
Conformance to the design criteria – if it doesn’t meet the design criteria, it is to be discarded. see upcoming posts regarding my clothing design criteria.
Quality – Aside from ranking the garments within a category, the quality of the garment also helps to group it into sub categories within a given style. More on this later.
Comfort
So, once the sorting is done, I discard those items below the line. Now, discarding doesn’t necessarily mean throwing out. In my wardrobe, clothing gets cycled down through various levels, until they eventually reach the bin. I call this the “hierarchy of chucking”
Wedding shirts, etc
Good going out clothes
Work clothes
Casual clothes
Lifeline \ op-shop
Around the house clothes
Exercise\yard-work clothes
Salvaged\upcycled to make something new
Rags
Rubbish bin
Now, obviously not every type of clothing can existing in each of the levels listed above, so they may skip a level, or start their life lower on the list.
So, I’ve rationalised my wardrobe to more sensible quantities and discarded the excess. as a nice little side bonus, K-mart also happened to have some storage tubs with lids that fit PERFECTLY into the top of my wardrobe, for things like cold weather gear, exercise equipment, etc.
That’s really all I had to say, but please look out for my upcoming blog posts about the correct criteria (at least, MY criteria) for selecting:
Tesla and I 