Strictly hypothetically speaking, let’s say a … ahem… friend of mine has seven pairs of Converse All-Stars Chuck Taylor high-tops in different exceptionally bright colors. Please do not judge at this point.
Let’s posit that this Chuck-obsessed person never wants to wear these beauties in mated pairs. After all, orange on both feet might feel pedestrian. But in the interest of fairness, for state occasions and under extreme peer pressure, matching shoes would be considered as a last resort.
Here’s the part where you help me… uh, I mean… this person crunch the numbers. How many combinations are possible:
- If the least restrictions are applied… any pairings
- If the pairs are never matched
- If you don’t count mirror image pairs (for example Pink left/Yellow right and Yellow left/Pink right are counted as one combination)
Please do not belittle my computational skills. Where crochet is concerned I can usually wrap my brain around most number problems. If the stitch repeat is 3.75 inches wide, how many repeats should be created in order to achieve garment sizes from XS to 3XL? What if there must be 2″ positive ease and no partial repeats? What if the number of repeats must be a multiple of 2? A multiple of 4? That I can do.
But I admit that I so suck at sneaker math it’s not funny and my head hurts. This Chuck problem keeps going round and round. Is there some elegant formula that gives me the magic number? Short of pulling out all seven pairs and lining them up and counting, I am totally confused here. Please, I need some sleep.
you have 7 colors and 2 feet so to get the total number of combinations possible: 7^2 (ie 7*7) = 49 combinations;
if the pairs are never matched: 49 total – 7 matching pairs = 42 combinations;
if you don’t count mirrors but do count matches then: 7+6+5+4+3+2+1 = 28 combinations. Think of it this way: You take the Pink Left and match it to every Right color for 7 combinations. Then you take the Yellow Left and match it to every Right color except Pink (because you’ve already done that one) so that’s 6 combinations. Then the Orange Left is matched it to every Right one except Pink and Yellow (which you’ve already done) so that’s 5 combinations. Etc.
I thought it was 21 because when you take the first color pink and match it to the other shoe there are only 6 possibilities not 7 because the pink/pink is out. So 6+5+4+3+2+1=21.
Jeane and Karla are both right. It just depends whether in the last question you allow matches or not.
Another way to think about the last problem is that you are essentially double counting the pairs, so as you said, Doris, pink yellow is the same as yellow pink. This means that you would take the number of ways to make a pair and divide by 2 (number of ways each pair could be arranged). So in the case of not allowing matches you get 42/2=21 possibilities. If you allow matches, this is slightly more complicated: you have the 21 nonmatched pairs plus your 7 matched pairs to give you 21+7=28 possibilities.
Hi! It made my head hurt for a minute too, and then I discovered I had an excuse to play with graph paper. (I like school supply shopping too, but I still have to do it with the kids.) I’ll try to recreate the chart I made here:
* R O Y G B I V
R M 1 2 3 4 5 6
O X M 7 8 9 10 11
Y X X M 12 13 14 15
G X X X M 16 17 18
B X X X X M 19 20
I X X X X X M 21
V X X X X X X M
Your total number of combinations is simply 7 squared, or 49. That’s your elegant formula. If you take out the 7 matched pairs (noted with M in the chart), you have 42 combinations- which makes our geeky little hearts happy. If you take out the mirror image of these combinations (marked with X in the chart), you cut that number in half, for 21 combinations (marked with their numbers). Adding back in the 7 matched pairs would give you 28 combinations. I am glad for the graph paper, I couldn’t see it in my head!
To sum up:
Any pairings= 49
No matches, yes mirrors= 42
No mirrors, yes matches= 28
No matches or mirrors= 21
I love fun shoes too, having once had a pair of plaid Converses (though they weren’t high tops). There are not enough fun shoes for women! So enjoy, and have a good sleep! 🙂
Don’t worry, Doris! Combinatorics doesn’t come easy, but once you get the hang of it, it can be fun. 🙂
I’m not certain of the why here … who cares about the math when you have all those sneakers? Just enjoy the sneaker ride. 🙂
How about putting different color laces in each of the shoes so you have even MORE combinations! i.e. right Pink with green lace + left Orange with blue lace, etc, etc, etc (hehehehe) Have fun 😉
Now my head hurts.
21 combinations with no mirror or matching pairs.
I’m glad all these lovely people did the math because that leaves me more time to contemplate the awesomeness of your Chuck Taylors.
To simplify things – you have 7 choices for the first of the pair, and 6 choices for the second of the pair. 7 times 6 = 42. That’s how many choices you have. Simple math!
7 pair shoes in different colors
If the least restrictions are applied… any pairings
N=7 colors^7 colors
If the pairs are never matched (no Red-Red)
N=7 colors*(7 colors-1 matching color)
If you don’t count mirror image pairs (for example Pink left/Yellow right and Yellow left/Pink right are counted as one combination)
N=(7colors-1matching color)+(6 colors-1matching color)+ (5 colors-1matching color)+4colors-1matching color)+3colors-1matching color)+2 colors-1matching color)+1 color-1matching color)
OMG! What a miracle of NERDS! Combinations, permutations, yarn, yarn, yarn…I have found me a home!
If it’s really only about the color combination, I agree with the 21. But then, of course, it makes a huge difference whether you’re wearing the green on your left foot or your right. So add the position into the calculation and you’ve instantly doubled the possibilities, making it 42. I’m seriously considering a copy-cat project, maybe starting off with a light version, just two colors. And then increase slowly… into the universe!
The first person is correct. There is an actual mathematical symbol used for this calculation. It is actually 7! if you are not counting mirrors, 2*7! if you do count mirrors where the exclamation mark indicates to take the number preceding it and add itself to every number preceding it (in this case 7+6+5+4+3+2+1). So not counting mirrors you have 7!=28 combinations, counting mirrors is 2*7!=56 combinations.
The answer to the first question is 49. 7 left shoes x 7 right shoes = 49 pairs
The second answer is 42. Since there are 7 ways to have pairs with the same colour, you remove those 7 pairs. 49 – 7 = 42 pairs.
The third answer depends on whether you add the pairs that match. Since the 42 pairs contain 2 possibilities of each colour combination (meaning a yellow left and a red right along with a red left and a yellow right), you divide those 42 pairs by 2. 42/2 = 21 pairs that don’t match. If you add those that do match, then it’s 21 +7 or 28. So, the answer for the last question is either 21 or 28 depending on if you add back the matching pairs.
I wonder if mathematicians and math teachers (I, included) find a certain appeal to crocheting/knitting.