#MathProblemOfTheDay: Halloween Costume Puzzle

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Remains of my last minute Halloween costume.

I dressed up as a puzzle. I taped numbers all over my pants. They were sorted into 2 clumps: one clump on each leg.  All the numbers on my left leg had something in common, and all the numbers on my right leg had something in common.

Hint: none of these numbers is divisible by 3.

Can you figure out why the numbers are sorted this way?

#MathProblemOfTheDay: Sitting at a Circular Table

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#MathProblemOfTheDay

Four people are gonna have lunch at a circular table. There’s an umbrella pole at the center of the table. How might you arrange the chairs so that everyone can see everyone?
PS: The arrangement pictured, where each chair is 90 degrees away from its neighbors, does not work, because people across from each other have look through the center to see each other.
PPS: one solution is just to put everyone right next to each other, leaving a whole half of the table vacated. Then no one has to look through the pole to see anyone else. But are there any solutions that distribute the 4 people around the table more evenly, minimizing the portion of the circle that is left vacant?

Original instagram post: https://instagram.com/p/3Mt8H0QJw0/

#MathProblemOfTheDay: Nike Running App

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Help me figure out how fast I ran!

I was gonna go for a walk and then decided to turn it into a yog. My routine was: walk a half mile, run a half mile. Repeat 4 times.

Nike Running tells me my average pace was a 14:17 minute mile. No magic there – they just took my total time (57:11) and divided it by my total distance (4 miles).

Anyway, here’s what happened. While I was walking I was relaxed. I was able to look at the screen and see my pace – about 19 minutes per mile. While I was running I had no energy or focus to spare – and I didn’t look at the screen. I want to find out roughly how fast I was running.

Given that my walking pace hovered around 19 minutes per mile, what was my average running pace?

#MathProblemOfTheDay: Skateparks and Concavity

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Skateparks are a great place to observe concavity.

A curve is concave up if its slope is increasing. This could mean the curve has a positive slope and it’s getting steeper, or it has a negative slope and it’s getting flatter. In both cases, we say the slope is increasing because it’s getting more positive. (Think about how -10 –> -7 is an increase just the way 7 –> 10 is.)

A curve is concave down if the opposite is true: the slope is decreasing. Either it has a positive slope and it’s getting flatter…. or it has a negative slope and it’s getting steeper.

Visually, you can think of a positive slope as an uphill, and a negative slope as a downhill.

A point of inflection is where the concavity changes from down to up, or from up to down.

Where is the point of inflection on this skateboard ramp?

P.S.  What would be the concavity of a straight line?

#MathProblemOfTheDay: keeping warm in the snow

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Which position would keep you warmer?

The answer is probably intuitive to you. That’s why I love this image – it connects simplicity and intuition to a more complex topic: optimization.

Optimization is an example of applied calculus. It sounds all fancy but it can be expressed simply. You want to optimize one thing (in other words, maximize something you want, or minimize something you don’t want), while keeping another thing constant (this is called a constraint).

In this photo, I want to minimize surface area to keep warm. (Why will minimizing surface area keep me warm??) My constraint is my volume – I can’t change the size of my body. It stays constant no matter how I position myself.

Ok, now imagine we’re no longer dealing with the human body. Take a piece of clay. What would you mold it into, in order to minimize surface area? How do you know you’ve succeeded?

#MathProblemOfTheDay: Slopes in a Subway Station

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This is part of the subway stop at 59th street and Lexington Ave, in New York City.

Estimate the slope of this escalator, by comparing the diagonal line to the grid of square tiles on the wall.

I found this interesting for two reasons:

(1) it seemed fun to consider different visual approaches here, and

(2) in real life, the staircase and adjacent escalator seemed *really steep.*  Yet if you look at boxes, the slope is less than 1, ie the angle of elevation is less than 45 degrees (a slope of 1 would be 1 full box up for every box over).

I remember in math class, I thought of a slope of 1 as kind of average. Not too flat, not too steep. But in the physical world it is pretty steep! I’m trying to think if I’ve ever seen a road, walkway, or ramp that actually had a slope of 1.

#MathProblemOfTheDay: Interesting Combination Lock

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I saw this combination lock at a Duane Reade.  It has only four options: you move the knob up, down, left, or right. They advertise that it’s easier to operate, especially if you need to open it in the dark.

However, it also seems easier for someone else to crack the code.

Assuming a 4-step combination code, how many tries would it take a stranger–at most–to crack the code? (I didn’t read exactly how many steps the code is, I just blindly assumed it would be 4). Assuming it would take 5 seconds to try a given combination, how many minutes, at most, would it take someone to bust open your lock?

You could also consider the following problem: suppose there is no fixed length of the combination. You can make any combination between 1 step long and 4 steps long. In that case, how many possible combos would exist?

Also: see what happens if the combination is 5 steps long.  Or 6 steps long.  The number of possible combos is shooting up.

Can you make a formula for the number of possible combos, if the combination is n steps long?