Well, most of us are not us lucky to fit into our wedding outfits way past our wedding year. Take this man. An important occasion demanded him to find this super white Italian shirt bought years ago for his wedding. Size 16. When he bravely tried it on, his wife noted that the collar suffocates him and he needs about 3mm more space everywhere in-between his neck and the collar all around his neck. What size shirts do you think should this man buy now?

Do you know that the shirt size 16 means that a tape wrapped around your neck and two fingers positioned flat upfront will measure 16 inches.

Image by jeremybrooks, distributed under CCL.

Answers accepted all day long on Friday April 29th and Saturday April 30th, on our Family Puzzle Marathon. They will be hidden until Sunday morning (EST) and everyone who solved it will get a puzzle point. Please, explain your answer.

## 13 comments:

For simplicity, let's assume that the collar wrapping around the neck takes a circular shape. The radius of the circle in inches and the size in inches are connected by this equation:

2*PI*Radius = Size. So for Size 16, it is: 2*PI*Radius = 16. From that, Radius = 16/(2*PI) = ~ 2.55 in.

3mm is approximately .12 of an inch (conversion factor being 25.4). So the appropriate size would have to have the LargerRadius of at least (Radius + 3mm) = 2.55 + .12 = 2.67in.

So the other size can be computed as 2*PI*LargerRadius = ~16.78. This is clearly more than 16 and a half, so they have to go for size 17 shirt. I hope the wedding was fun!

I get 16.85" as the updated circumference, so that probably makes the shirt size 17. Solution: 16 is the current circumference so the diameter is 4.51" based on circumference = pi x r squared. He needs 3mm more diameter which is 3/25.4 = .1181" (25.4 mm per inch). That makes the new circumference 16.85"

Allen Cohen

If the wife is correct and he "needs" exactly 3mm more added to the radius, that'd be 6mm added diameter or almost 19 mm= .748 inch more circumference, for the fit that's being prescribed. Get a 17" collar next. 17.5" might last him longer. Try it on. Duh.

Experience tells me that his 16" collars will not fit him again for a very long time, if he's going to button them. I haven't worn a tie for 4 years (retired) so I continue to use up some old shirts that should probably be given away.

So a size 16 shirt is a circumference of 16 inches. With a need for an additional 3mm all around, I added 3mm to the radius of the circle and calculated the new circumference to be 425.6mm. That's between 16.5 and 17 inches, so I think he needs to buy a new shirt in size 17 to accomodate his neck now - the 16 1/2 would be too small.

This is my answer to the puzzle. Since the collar is tight, I assume that he has no extra room and that he needs the extra 3 mm. 3 mm equals 0.12 inches. From my experience, shirt sizes come in half sizes. 16 1/2 should give him more than enough room. He would have about .38 inch left to spare.

GB

I'm going to assume I can treat his neck as circular.

So. A size 16" circumference = pi*2*r. If I find r, convert to mm, add the 3mm he needs all around, I should be able to use my new r to get the new circumference.

r = 8" / pi = 2.5465" = 64.6811 mm

So the new radius is 67.6811 mm, making the circumference 67.6811 * pi * 2 = 425.2525 mm.

Convert to inches = 16.72 inches.

It looks like he'll need a size 17 shirt to avoid choking.

Alternatively, a second method:

Wedding era 2*pi*r=16. The current circumference = 2*pi*(r+3mm). Covert the mm to inches and distribute: 2*pi*r + 2*pi*.118 inches = neck size. We know 2*pi*r = 16, so the new circumference is 16+2*pi*.118 = 16.74 inches.

Close enough. He'd still better get the 17.

Knowing that the circumference of a circle is equal to 2r(pi). With this we can calculate the radius of the circle created by the size 16 shirt to be 2.5465 inches using simple Algebra.

His wife is recommending that he increase radius by 3mm. Converting the 3mm into inches give you 0.1181 inches. The new radius of the shirt that would fit needs to be 2.6646. Using the formula 2r(pi) again, and we have a circumference (or shirt size) of 16.742 inches. Since short sizes are in 1/4 inches, he should get a 16 3/4 and it should fit fine.

Carrie

I used the circle circumference formula even though the neck circumference is not a circle.

We have 2*PI* R=16

i added (3*0.04)to the radius I found and then I solved for the new circumference and I found 16.75

Anne-Marie.

pi*d = circumference of a circle.

For the shirt we have

pi*d = 16 inches = old circumference of his neck

His wife determines he needs an additional 6mm for the diameter (3mm all the way around), but we will have to convert that to inches.

new circumference = pi*(d + 6mm*(1cm/10mm)*(1 in/2.54cm))

= pi*d + pi*6/25.4 in

= 16 in + 0.2362*pi in

= 16.74 in

Since shirt sizes go up in half-inch increments, he should buy a size 17.

The guy needs a shirt with a 16.75 inch neck. His current shirt has a circumference of 16 inches but is too tight. We need to find the diameter and add 6mm. 16 divided by 3.14(pi) is 5.095 inches. I then converted 6mm. to .24 of an inch. I then added .24 inch to 5.095 inch and got 5.335 inches for a diameter. Finally, I multiplied 5.335 by pi and got 16.75 inches.

Since its an Italian shirt, I also found an answer in cm. I converted the 5.095 inch diameter to 12.94 cm. and added .6 cm. to get a diameter of 13.54 cm. Then I multiplied the diameter by 3.14 (pi) and got 42.52 cm. mathmover

We can equate the collar encircling the neck to a circle. So, because the original shirt size is 16, that means that the circumference of the shirt collar is 16 inches [circumference = pi (3.14) x diameter]. The diameter of a 16 inch shirt then works out to be 5.0955 inches.

If we are to add 3mm more space everywhere, that would add 6 mm to the diameter of the circle [3 mm on each side]. 6 mm works out to be 0.2362 inches which we then add to the 5.0955 original shirt diameter, giving us 5.3317 inches for the larger shirt collar diameter. This diameter leads us to a circumference of 16.7415 inches. I'm assuming that shirt sizes possibly come in whole and half sizes but that you would want to go up to the closest size to a 17 inch circumference or size 17 shirt.

Next you're going to ask us how many shirt and tie combinations we can find to wear to this important event...possibly the royal wedding? Oh, and don't forget the hat ;<)

Look at you all - fashion experts! Almost all of you came up with an answer around 16.75" and suggestion for a size 17 shirt. The explanations above are so accurate that I won't repeat them again.

2 x Pi x R = 16

2 x Pi x (R+ 0.0018) = ? = 16.75

A puzzle point for Ilya, Allen, Tom, PApaige, GB, Bean, Carrie, Anne-Marie, kj, mathmover, thelittlebird.

mathmover - please shoot me an email. I run out of the mathmom mugs and now will be sending Tshirts as 5 puzzle points prizes. I need to know your size :)

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