If you have a square box filled with two layers of chocolate squares and you eat all 20 pieces in the upper layer along the wall, how many are left in the box? Is there a graphical way to set this up or a math equation? Not only what is the answer and how do you get it, but how do you approach problems like this from a mathematician's mindset?

It's not great wording, but I think this is the idea:1. The box is square. This means there will be the same number of pieces on each side of the box.2. The two layers are identical.3. The 20 pieces she ate were all pieces that touched a wall of the box.Draw a picture-- in a 2 x 2 box there are 4 edge pieces and no center pieces.In a 3 x 3 box there are 8 edge pieces and 1 center piece.In a 4 x 4 box there are 12 edge pieces and 4 center pieces.In a 5 x 5 box there are 16 edge pieces and 9 center pieces.In a 6 x 6 box there are 20 edge pieces and 16 center pieces. That's the one we want-- it has 20 edge pieces.So the box is 6 x 6 x 2 layers, or holds 72 pieces. If she has eaten 20, 52 are left.Having written out the various possibilities, it becomes clear that one way to solve this for very large numbers of pieces is to write out the formulaedge_pieces = side_length^2 - (side_length-2)^2put in 20 for edge pieces, and solve for side length, which you can use to figure out the number of pieces left. But most mathematicians would probably draw a few examples in order to work out the formula anyway.

Being a girl, if you have to get out of your home for the first time with girls who you don't know and they have already formed groups of their own and you are just alone, how do you stay independent in this entire office trip of a week or two?

Your age is not really clear.I assume you are an adult, not a girl, since you are going on an office trip.I wouldn’t worry about it, it is not a social trip, so you don’t have to have friends around you.If you want to be independent, do some research on the destination and find some activities and attractions nearby you can check out during the free time in your schedule.But there are usually more than one person who doesn’t buddy up with coworkers. You may find others looking for company on the trip.

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If you have a square box filled with two layers of chocolate squares and you eat all 20 pieces in the upper layer along the wall, how many are left in the box? Is there a graphical way to set this up or a math equation? Not only what is the answer and how do you get it, but how do you approach problems like this from a mathematician's mindset?
It's not great wording, but I think this is the idea:1. The box is square. This means there will be the same number of pieces on each side of the box.2. The two layers are identical.3. The 20 pieces she ate were all pieces that touched a wall of the box.Draw a picture-- in a 2 x 2 box there are 4 edge pieces and no center pieces.In a 3 x 3 box there are 8 edge pieces and 1 center piece.In a 4 x 4 box there are 12 edge pieces and 4 center pieces.In a 5 x 5 box there are 16 edge pieces and 9 center pieces.In a 6 x 6 box there are 20 edge pieces and 16 center pieces. That's the one we want-- it has 20 edge pieces.So the box is 6 x 6 x 2 layers, or holds 72 pieces. If she has eaten 20, 52 are left.Having written out the various possibilities, it becomes clear that one way to solve this for very large numbers of pieces is to write out the formulaedge_pieces = side_length^2 - (side_length-2)^2put in 20 for edge pieces, and solve for side length, which you can use to figure out the number of pieces left. But most mathematicians would probably draw a few examples in order to work out the formula anyway.
Being a girl, if you have to get out of your home for the first time with girls who you don't know and they have already formed groups of their own and you are just alone, how do you stay independent in this entire office trip of a week or two?
Your age is not really clear.I assume you are an adult, not a girl, since you are going on an office trip.I wouldn’t worry about it, it is not a social trip, so you don’t have to have friends around you.If you want to be independent, do some research on the destination and find some activities and attractions nearby you can check out during the free time in your schedule.But there are usually more than one person who doesn’t buddy up with coworkers. You may find others looking for company on the trip.

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