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Check out the past discussion problems below. Make sure to scroll right to see the entire collection.
(You need to view this on a laptop)
Join our discord server to join the discussion of these problems.
Our puzzles are based on public resources provided by JRMF.org.
Tower of Hanoi
In class, we talked about how the Tower of Hanoi is based on recursive relations. We also found the recursive relation for the Tower of Hanoi with three pegs. We then explored the Tower of Hanoi with two rings of each color, four pegs, and five pegs. We ended by exploring the Tower of Hanoi with colored pegs. In particular, we explored the case where A and B are pegs of one color, while C is another color. You can only move rings between pegs of different colors. Can you find the number of moves with four rings and five rings for the Tower of Hanoi with colored pegs?
River Crossings
In class, we started with the classic river crossing problem of a wolf, a goat, and a cabbage. We then moved on to a variation in which we need to carry humans and zombies instead. Remember, if there are more zombies than humans on any side of the river, all of the humans on that side will die. How would you solve the case with 4 humans and 4 zombies, along with a boat with two slots? Is it impossible? If so, what is your reasoning?
Chameleons
Last class we learned about the conditions we need to make all three colors of chameleons into one color, which was having the same amount of two colors of chameleons. Can you figure out the pattern needed to make this condition happen, and in turn, the conditions for being able to turn all fifteen chameleons into a single color?
Königsberg
In class, we started with the classic Königsberg problem of drawing a shape with one continuous line. The shape is made up of nodes and lines connected to them; you can go over the nodes multiple times but only once for the lines. We practiced numerous questions before explaining how to identify whether a shape is crossable or not. Lastly, we provided real-world problems that use the same application as the Königsberg problem. What is the most efficient way to go through all of the lines? What additional lines do you need to add if the shape is not crossable?
Cup Stacking
This week, we talked about the Cup Stacking problem. We can solve the 3x3 problem by stacking all of the cups in the middle. For the 5x5 problem, we can solve it similarly by focusing on the corners first. Can you find a way to apply these methods to the 7x7 and 9x9 problems? (Hint: For the 9x9 problem, solving the corners will leave leftovers. How can you split the leftovers evenly to move to the center?
Jumping Julia
This class, we explored the Jumping Julia puzzle. Look at the puzzle attached. Is this puzzle possible? Using the working backwards strategy we used in class, what is the shortest path to the goal? Which squares are impossible to reach? Is there a fast way to find all of them?
Domino Dissection
In addition to some interesting AMC 8 problems, we solved some Domino Dissection puzzles. Puzzle 190 is a very hard 8x7 puzzle, see if you can solve it!
Colored Loops
This week, we learned about the Pythagorean Theorem and applied it to a variety of interesting problems. We also tried our hands at the Colored Loops puzzle. The puzzle in the picture has three solutions. Can you try to find all of the solutions? (hint: try starting on one color and finding a complete loop.)
Here is the Jamboard where you can access the problems we solved and the class notes: https://jamboard.google.com/d/1prdftFsEDVCOCEWnDnNdJzxaBMut4DMYNR3aczH3cbk/viewer?f=4
Make sure to let us know if you have any questions!
Here is the Jamboard where you can access the problems we solved and the class notes: https://jamboard.google.com/d/1prdftFsEDVCOCEWnDnNdJzxaBMut4DMYNR3aczH3cbk/viewer?f=4
Make sure to let us know if you have any questions!
Prime Cubes
After solving some AMC 8 problems involving special operations, we looked at the Prime Cubes puzzle. Recall how we were able to fill the last puzzle with all of the numbers. Going back to the first puzzle (the Prime Cube), is it possible to fill all 8 numbers?
Lot's o' Plots
In today's meeting, we talked about circles in the AMC 8 and learned how to prove mathematically which sizes of plots could be filled with carrots (1x2 blocks). Can you find a way to prove which plots can be filled with lettuce (1x3 blocks)? How is this different than the carrots other than the difference in area?
Here is the Jamboard where you can access the AMC 8 problems we discussed in class and the class notes: https://jamboard.google.com/d/1-WZSSGqYZWXd0f6SL--LyGwcz1VdnKr_ASyiD76QEZ8/viewer
Here is the Jamboard where you can access the AMC 8 problems we discussed in class and the class notes: https://jamboard.google.com/d/1-WZSSGqYZWXd0f6SL--LyGwcz1VdnKr_ASyiD76QEZ8/viewer
Crack the Code
In this class, we explored the puzzle Crack the Code. We saw that one excellent method to deduce correct colors is to start with a column of the same colors and vary one spot at the time for each guess. However, this method proves to be insufficient in a harder puzzle like the one shown above. Can you come up with a better method for solving these puzzles?
Sprouts
Last class we played the game of Sprouts. Given the number of starting points, can you find out the maximum and minimum number of moves for a game to end?
Geometric Construction: Perpendicular Lines and Polygons
This class, we explored the basics of geometric construction. We discovered how to construct a perpendicular bisector, an equilateral triangle, and a square. We also discovered a fancy way to construct a regular hexagon, as shown above. Using the techniques that we learned, can you find a way to construct a regular dodecagon (12 sides)?
Construction: Circumcircles and Incircles
In this class, we talked about incircles, circumcircles, and their construction. We also talked about how to construct an angle bisector. Can you construct an angle of measure 45 degrees? What about 15 degrees?
False Proofs
This class, we talked about the principles of false proofs. We went over several examples of false proofs and saw that two operations that result in false statements are dividing by 0 and carelessly taking square roots. Can you create your own false proof using either of the two operations stated?
Here is the jamboard with the examples from class: https://jamboard.google.com/d/1cxLVwmkYW7eSVHS0UFRPgp42OnnDJVM39arCaJFNRUw/viewer?f=4
Here is the jamboard with the examples from class: https://jamboard.google.com/d/1cxLVwmkYW7eSVHS0UFRPgp42OnnDJVM39arCaJFNRUw/viewer?f=4
Fruit Puzzles
In this class, we solved a variety of fruit puzzles. We also introduced algebraic systems of equations, including methods for solving them: substitution and elimination. Using what we learned in class, can you solve this fruit puzzle?
You can access the puzzles using this jamboard: https://jamboard.google.com/d/1FDIiJSSK8GG8gns7PRWpTmSmeoNAQshsjxImJfpLR5w/viewer?f=12
You can access the puzzles using this jamboard: https://jamboard.google.com/d/1FDIiJSSK8GG8gns7PRWpTmSmeoNAQshsjxImJfpLR5w/viewer?f=12
Square Puzzles
Today, we talked about square puzzles. We started out with basic ones involving pure numbers, then we explored square puzzles that involved variables. We used it as an opportunity to introduce systems of equations. Try this puzzle and hone your skills!
Basic Combinatorics
Today, we introduced factorials, permutations, and combinations as part of basic combinatorics. How many arrangements can you make from the word "TWO_WORDS"? The space in the middle matters! We are looking for two separate words, so the space cannot go in the front or back of all the letters.
As a bonus problem, figure out the number of rearrangements of the world's longest word, shown in the image above.
Here is the jamboard used in class: https://jamboard.google.com/d/1ZjbvNgUvHjqLtH5WADZs7SeoRBNuGhowX65lFnDTD3s/viewer?f=12
As a bonus problem, figure out the number of rearrangements of the world's longest word, shown in the image above.
Here is the jamboard used in class: https://jamboard.google.com/d/1ZjbvNgUvHjqLtH5WADZs7SeoRBNuGhowX65lFnDTD3s/viewer?f=12
Basic Probability
Today, we talked probability. We talked about the basic way of computing probabilities by dividing the number of successful outcomes by the number of total outcomes, and also how we can multiply and add probabilities. Lastly, we showed how we could solve problems by using complementary counting. Now that you are a master at probability, can you solve this problem from 1989 AJHSME (Problem 25)?
Every time these two wheels are spun, two numbers are selected by the pointers. What is the probability that the sum of the two selected numbers is even?
Every time these two wheels are spun, two numbers are selected by the pointers. What is the probability that the sum of the two selected numbers is even?
Pascal's Triangle
In class, we talked about Pascal's Triangle and the many formulas and identities that can be derived from it, one of it being the Binomial Theorem. Can you figure out the coefficient of x^2 in the expansion of (2x+1)^4?
Systems of Equations
Today, we went over the basic techniques of solving systems of equations: substitution, elimination, and isolation. Now that you are a master at solving systems of equations, give a go at this problem!
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