Brain Teasers ? or 50 Years On........... ?

Each of the three different symbols above has a different value associated with it. Adding up the value of the symbols in each row and column gives you the value for that row or column. What is the value of each symbol and what is the missing value of the column ?

This, like the previous one, is just a "Warm-up" for the "real" one shortly

Man = 4 , Case = 21 , Plane = 8

Missing row = 41

I saw this problem at fivethirtyeight.org. It's called the "Picky Eater Problem":

A man likes sandwiches, but he hates crusts. In fact, he hates bread crusts so much that he will only eat halfway to the crust from the center of the sandwich. Given a square sandwich, what fraction of the sandwich will he eat?

I think I know what the eaten portion will look like, but I'd be interested in hearing what other people come up with. The answer hasn't been published yet.

1/4.

Egg sandwich. Egg extending only halfway to the crust.

Looks like I agree with Frank.

A paper circle with a radius of 2 (D=4) is folded so that the circumference touches (and is tangential with) the diameter 1/3D of the way from the circumference (measured along the diameter). What is the length of the fold?

I disagree with both answers to the sandwich problem. It's the distance from the center point of the square, not a center line. The corners of the yellow square are a lot farther from the center point than they are from the closest point on an edge.

JRHardee posted:

I disagree with both answers to the sandwich problem. It's the distance from the center point of the square, not a center line. The corners of the yellow square are a lot farther from the center point than they are from an edge.

I think you're wrong. When he's eating in the diagonal direction, the point half-way to the edge is as shown in the sketch. There seems to be no other interpretation that makes sense. If the direction (line) of eating and measurement aren't constrained to be the same, there are infinite solutions.

I edited my response to be more clear about distances. I agree that the yellow corner is halfway to the corner of the sandwich, but points on the edge of the sandwich are much closer than the corner. He won't eat as far as the yellow corner.

winkyincanada posted:

JRHardee posted:

I disagree with both answers to the sandwich problem. It's the distance from the center point of the square, not a center line. The corners of the yellow square are a lot farther from the center point than they are from an edge.

I think you might be right or wrong. When he's eating in the diagonal direction, the point half-way to the edge (in this case, the corner) is as shown in the sketch. If the direction (line) of eating and measurement aren't constrained to be the same, there may be infinite solutions. One could interpret the "distance to the edge" to always be the shortest distance (i.e. to the nearest edge). That solution seems much more complex. I'm imagining a similar square but perhaps rotated 45 degrees and smaller. Or an octagon?

No--you are indeed halfway to the corner, but you are more than halfway to the closest point on an edge.

I get an answer of 0.2203

This is the method I used:

1) I split the square into 8 half quadrants. If the square length is 4R then the distance from the square centre to the edge is 2R and the safe eating distance to the edge is R

2) Moving towards the corner then the radial safe eating distance is R / (cos Theta)  where Theta is the angle subtended at the centre

3) You now need to work out the area of the "sector" as you move go from Theta =0 to Theta = 45 degrees. 

4) This means integrating 1/ cos Theta between 0 and 45 degrees. The integral of 1/ cos Theta = ln (sec Theta + tan Theta)

5) The total safe eating area is 8 times the sector area. I get 3.5249 * R^2

6) The total area of the bread (with crusts) is 16 R^2

7) Total fraction of safe bread is 3.5249/16 = 0.2203

Regards,

Peter

JRHardee posted:

No--you are indeed halfway to the corner, but you are more than halfway to the closest point on an edge.

What Winky said.

 “eat halfway to the crust from the center” This must mean that he is eating in a straight line from the centre towards the crust. It can’t mean anything else.

 Lets say your interpretation is correct, if he starts eating towards the top left corner, how can he ever be half way towards the bottom right corner.

My thought was to draw a square, then draw the diagonals. For each of the resulting triangles, the problem reduces to finding the locus of points equidistant from a point (the center point of the sandwich) and a line (the crust), which is a parabola. You get a four-sided trefoil type of figure made of four partial parabolas. I haven't done a definite integration since the Nixon Administration, so I leave that to others.

I've just had a look on the fivethirtyeight website and spotted a similar puzzle.

Every morning, before heading to work, you make a sandwich for lunch using perfectly square bread. But you hate the crust. You hate the crust so much that you’ll only eat the portion of the sandwich that is closer to its center than to its edges so that you don’t run the risk of accidentally biting down on that charred, stiff perimeter. How much of the sandwich will you eat?

I can see Sophies 8 half quadrant approach being the solution for the above.

The very same--as I mentioned. Still waiting for a published solution.

(Egg) sandwiches………and at the risk of getting egg on my face

The sandwich is considered as 8 triangles ABC ( I am adopting Peter/Sophie's concept above)

• Angles A and C are 45 deg
• Angle B is 90 deg
• AB = BC = 1 (I am being a Plonker and doing some crude numerical evaluations !)
• AC = 1.414214

• M is the mid-point of AC
• N is the mid-point of BC

• We are looking for the locus of N as it moves to M
• (yes, I know it looks like a straight line, but………let’s find out )

I intend to calculate the ordinate (the “vertical” distance from MN to the horizontal line through the sandwich) of the radials at 0 deg; 15 deg; 30 deg and 45 deg measured anti-clockwise from BC. And I will chuck in 27 deg as a “random” for good measure.

If necessary, I can then use Simpson’s rule to estimate the area of the sandwich between the locus MN and the line AC. (multiply it by 8 and “Bingo” )

Now, before I do these calculations, am I on the same wavelength as everybody else, or have I fundamentally mis-understood the problem ? [Yes you Plonker, I hear you say ]

Picture to follow !

Don Atkinson posted:

Now, before I do these calculations, am I on the same wavelength as everybody else, or have I fundamentally mis-understood the problem ? [Yes you Plonker, I hear you say ]

 

Don,

I don't think the locus will hit point M. The distance CM is greater than the distance from M to the line AB.

The point along the line AC, (lets call it X) is when the distance CX equals the shortest distance from X to the line AB. possibly.

Don

I think we are on the same wavelength (very dangerous !)

I believe the resulting shape just adds a little convex curvature to the line NM in your diagram. If you just take the area of the triangle MCH you get an area of 0433 * R^2. By using the integration method which I explained (so poorly) in my original response, I get an area for the "sector" as 0.44068 * R^2.

Look for to hearing what Mr Simpson has to say :-)

Regards,

Peter

OK, thanks to Frank and Peter (and others) I think i've got it now.

The "horizontal" line MN in my diagram is not parallel to the crust AB.

It curves gently from N (where it is parallel to AB) toward the diagonal AC which it does cut at 45 deg, (but at a point closer to C than where M is shown)

The locus ("X" using Frank's nomenclature) is defined by two lines of equal length. The line CX and the "Vertical" line from X to AB. (Frank used the more accurate term "shortest distance")

In my defence I would point out that I did say (and I quote !!!!!!) - (yes, I know it looks like a straight line, but………let’s find out )

Cheers, Don (Clutching at straws)

Does anybody else have a similar picture in mind when contemplating winky's puzzle ?

Don Atkinson posted:

Does anybody else have a similar picture in mind when contemplating winky's puzzle ?

Yep, that is it.

Winky

I get a fold line length of 2.91 units. But I might well have made some errors along the way !

I just solved for the intersection points of the 2 circles and then used Pythagoras to work out the fold lengh.

Regards,

Peter