BMAD

I stand corrected. I didn't catch that i rounded :b

A partition of a positive integer n is a way if writing n as a sum of positive integers, ignoring the order of the summands. For example, a partition of 7 is 3 + 2 + 1 + 1. The table below shows all partitions of 5. The number of 1s column shows how many times the number 1 occurs in each partition. The number of distinct parts column shows how many distinct numbers occur in each partition. The sum for each column, over all the partitions of 5, is shown at the foot of the table. (attached) Let a(n) be the number of 1s in all the partitions of n. Let b(n) be the sum, over all partitions of n, of the number of distinct parts. The above table demonstrates that a(5) = b(5). Show that, for all n, a(n) = b(n).

That is a nice improvement CaptainEd but still even that isn't the maximum distance.

It always amazes me when people use strategies hidden in the problem that i didn't even consider and even more makes me jealous when their strategy is more efficient than my own.

I would think that it depends if the clone also had memories and understandings cloned with it. Does the clone 'remember' and 'know' what they have experienced and do others treat this clone as the original or as a clone of the original. If all are in the affirmative then i don't see why not.

A triangle has sides 10, 17, and 21. A square is inscribed in the triangle. One side of the square lies on the longest side of the triangle. The other two vertices of the square touch the two shorter sides of the triangle. What is the length of the side of the square?

The towns of Alpha, Beta, and Gamma are equidistant from each other. If a car is three miles from Alpha and four miles from Beta, what is the maximum possible distance of the car from Gamma? Assume the land is flat.

It depends on who is doing the showing. If aliens 2000 light years away have shown the knight the state of their electoral procedure 1999 years in the future then there is nothing the knight can do change that process as any communication the knight has sent a message to indicate that senator Palpatine is in fact Sith will arrive 1 year too late. No one can stop the dark side. From our viewpoint the knights view of the future has already happened. If however we did the showing because we are not 2000 light years away from the knight, provided paradox does not ensue Hitler wins World War II after experiments in cloning. I am unsure on this... if the knight perceives an event further away then could be arrived (ie. 2000 light years away from an event when <2000 light years from when the event started) then i still think a chaos event could still happen....here me out...If knight perceives an event, said event changes knights society, said society voyages to alien soiety and arrives well after the event has happened but informs alien race with astonishing accuracy the details of their historic election. All the while, the knight (or his decendents) are watching alien and human interaction based off his original observation.

I don't think even this idea would be safe, as the showing still changes the attitudes of the knight you could bring about the concept of democracy early in a feudal based society. This in turn could affect the prime ministers campaign and subsequent election. The only way to be sure that a small change in the past does not mess up the future indicated is to make sure that the two time periods cannot be related at all. If instead you showed the knight the election of the alien prime minister of a planet as at most many years in the future as light years away it might work. This is dependant on no causal relationship being able to travel faster than light Even then we would have to make sure the knight couldn't perceive or understand what they are seeing. It needs to be not only alien but incomprehensible.

The January of certain year had four Fridays and four Mondays. What day of the week was January 20th of that year?

This one may be too easy, but enjoy anyways (see attached).

Given that triangles have three sides and three angles we could share this information to another person to sketch our triangle. Of course, it is a known fact that we only would need to share three parts (sides/angles) at a minimum with someone else in order for them to construct our shape (e.g. Side, Side, and Side; Side, angle, and side). It is also known that to have someone build a quadrilateral for us, it would require five pieces of information. But what about a pentagon, hexagon,..., octagon, ... n-agon? Is there a generalizable formula as to the minimum amount of information needed for someone to construct your polygon unseen? Shape: Minimum Information Needed: Triangle 3 quadrilateral 5 Pentagon Hexagon ..... Octagon n-agon

2 comments. 1. Why do most take the negative view? What would happen if the future machine show a positive event? A future marriage, wealth enhancement etc. would that only lead to a good day as you know it is going to work out for you? Morality aside: 2. The problem, IMO, lies in the fact of what Phaze said, since the machine must look at every variable outside of itself then that must include the idea that the machine must also assume that the person themselves has no knowledge of the future as well and so, providing the individual with this knowledge introduces a variable that it has not calculated before meaning that the prediction it calculated prior to showing the future to the user is based on a different set of equations and thus is fundamentally not predicting the future of the person that knows. Now the way around this. I believe is a future machine that shows the future to a user of someone that the individual, even if they tried, could never impact. E.G. Showing a medieval knight the election of the prime minister of Malaysia in the year 2072 may be a safe.

There are two reasonable and acceptable answers.

And if you are a nifty folder, no tape or glue needed

I thought about that. But the OP states your child needs to "cut out 10 circles". Her assignment was not to "glue," "tape," "construct," or even "make" -- it was "cut out." Although, she should still get her credit and praise for creativity, if she does not blab out that a parent has done the project for her. How many circles (rings) would you be able to cut if you cut the paper diagonally?

(I don't understand how to load up images here, so see attached) Ignore my poor attempts at Microsoft Paint for a second and presume that the same colors produce congruent pieces. Problem: When a student of mine calculated the individual combined pieces they found the area to be 59 sq. units but when they put them together and use the area formula they, they found the area to be 60 sq units and again, when they changed the orientation of the pieces to make another version and used the area formula they found the area to be 58 sq. units. What's going on here?

I would argue that this question is a tricky question not a trick question. Too find the Range of yard for the dog you must consider ALL possibilities but you guys have only considered the house to be rectangular.

Sure. There are an infinite number of tie points. Do you want a general equation? Do you want the tie point for max or min? Something other? A simple range of values would suffice so that i know the potential roaming of my dog.