13 Math Problems to Start Off 2013

Because why not.

1) A particular alien warrior has eight hands, arranged generally in the directions of the vertices of a cube. When it goes to war, each hand must carry a sword, a shield or nothing, and it must carry at least one sword. This alien is rotationally symmetric over the sagittal-coronal axis, and the cube that the hands form has edges parallel to these and the transverse axis. Note that the alien is not reflectively symmetric over the transverse plane. In how many distinct ways can this alien arm?

2) Paula is usually a reliable painter, but once in a while, she does make a snafu. Suppose that on the nth day that Paula works, she accidentally spills φ(n)-2 gallons of paint, where φ(n) is the number of positive integers less than or equal to n that are relatively prime to n. Fortunately, due to the fact that her boss was impressed by her ability to spill antipaint that she demonstrated on her first two days at work, her boss decided to refrain from firing her until her work becomes unprofitable. Her work becomes unprofitable when the total number of gallons of paint she has spilled exceeds the number of days she has worked. After what day of her work does Paula get fired?

3) An RPG hero is on a quest to put an end to the rampant inflation plaguing the world. The first night, the stay at an inn costs 100 gollld. The second night, a stay at a commensurate inn costs 200 gollld. This rate of inflation persists until the end of the hero’s journey and for two days before its start. Before the thirty-second night of the journey, the source of inflation is finally vanquished. Assuming the prices immediately drop to their original values the next day, what is the annual rate of deflation of the value of gollld over that night?

4) In one of the journeys of the hero mentioned in the above problem, the hero encounters a locked door. The hero is a swordsman and can deal 200 HP of damage with one blow, which takes one second to execute. With the hero is an archer who can shoot an arrow which deals 250 HP of damage with one blow, which takes one and a half seconds to execute. Also in the party is a mage whose best spell deals 900 HP of damage and takes five seconds to execute. Assuming that if they don’t bring the door down within 60 seconds, they will get bored and actually look for the key, minimally how much HP must the door have to force the party to ascertain the location of its key?

5) Suppose a paper beats a rock if it can actually wrap all the way around a rock. Suppose paper has a density of 500 kilograms per cubic meter and rock has a density of 6000 kilograms per cubic meter. Also, suppose paper is 0.1 millimeters thick and rectangular and that rock is spherical. Is there a particular weight above or below which the rock requires a heavier amount of paper to be able to beat it, and what is this weight?

6) Two Asians have just finished dining, and have finally agreed to split the bill exactly in half and have each party pay their half. The bill was $27.83, so the agreement required the chopping of a penny. Although the one who was agreed on to do the chopping was pretty proficient at martial arts, he is unsuccessful at splitting the penny close enough to half three-fourth of the time, thus calling for the bringing out of a new penny to split in half. Each preparing and splitting of a penny takes ten seconds. Suppose that each minute is worth $1.00 to each of them. How long must they have been arguing over who pays the bill before they agreed on the method of paying for it for there to be as much money wasted then as is the expected amount of money wasted while attempting to split a penny in half?

7) Suppose in the above problem that one person plans to pay with a credit card as soon as the other is done successfully halving the penny, and the other person is repaying half of the bill over to the first person in cash. What is the expected number of minutes it must take for the restaurant to become cash only so that the first person will have to take back the credit card and find cash to pay with?

8) The author of these problems was too lazy to write thirteen math problems, and thus decided to call that the “13” in the title of this set of problems was written in another base. Fearing that his excuse was too base, he decided to neutralize it a bit. Suppose this excuse has a pH of 13 (base five) and is monoprotic. 20 mL of the excuse is present. One molar hydrochloric acid must be titrated with this excuse to reach neutrality. If it takes one second to dispense 5 milliliters of acid, how much shorter does it take to neutralize the excuse a bit rather than to neutralize it a byte, assuming that a byte is the full amount of neutralization?

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