Beginning of the year Problems

For the beginning of the year while waiting for a physical room to become available  I sent out this series of  problems on a daily basis as teasers for what we would do once we starting meeting.


#1  To start: try your hand at decoding the following message which was encoding with a substitution cipher.


TABGIWA VI VRA WLVR GBSM. U LW ANGUVAQ VI MA FVLYVUPH VRUF XALY. UO XIS GLP YALQ VRUF WAFFLHA FAPQ WA MLGJ XISY OLKIYUVA FPLGJ OIIQ. FAA XIS PANV TAAJ.


If you haven't tried cracking one of these before here's a good starting link:


Also don't forget to use a letter frequency chart:

By letterBy frequency
LetterFrequencyLetterFrequency
a0.08167e0.12702
b0.01492t0.09056
c0.02782a0.08167
d0.04253o0.07507
e0.12702i0.06966
f0.02228n0.06749
g0.02015s0.06327
h0.06094h0.06094
i0.06966r0.05987
j0.00153d0.04253
k0.00772l0.04025
l0.04025c0.02782
m0.02406u0.02758
n0.06749m0.02406
o0.07507w0.02360
p0.01929f0.02228
q0.00095g0.02015
r0.05987y0.01974
s0.06327p0.01929
t0.09056b0.01492
u0.02758v0.00978
v0.00978k0.00772
w0.02360j0.00153
x0.00150x0.00150
y0.01974q0.00095
z0.00074z0.00074

#2

Today's puzzle is geometry focused and comes from the Julia Robinson Festival.

A few notes about these type problems.

  * When finding areas, you almost always want to find simpler shapes like triangles or squares within more complex shapes.  Or another way of saying the same thing,  you probably will need to draw in some more lines.


  * One other thing to try is to make a physical model out of two pieces of paper and experiment if you're not immediately thinking of any ideas to explore.


#3 

The product of a 3-digit number and a 1-digit number is 3714. What is the 3-digit number?

#4


Without further ado, today's puzzle is a grid logic problem. These are usually best solved by formed a grid that matches each of the parameters you have to sort out:

Assassin is a popular game on college campuses. The game consists of several players trying to eliminate the others by means of squirting them with water pistols in order to be the last survivor. Once hit, the player is out of the game. Game play is fair play at all times and all locations, and tends to last several days depending on the number of participants and their stealth. At Troyhill University, 5 students participated in a game that only lasted four days. Can you determine each player's first name, their color, their assassin alias, how they were eliminated, and their major?

Names: Liam, Anabel, Bella, Oliver, Ethan
Colors: Red, Green, Blue, Purple, Black
Alias: Captain Dawn, Night Stalker, Dark Elf, McStealth, Billy
Capture: Caught at weekly study group, Caught helping friend with car trouble, Ambushed during sleep, Caught on the way to class, Winner
Major: Economics, Biology, Art History, Sociology, Psychology

MONDAY: Liam, the girl named Captain Dawn, and the person in purple avoided any action that day. The psychology major was able to easily catch Ethan because she already had a study group meeting with him that day. Since it was a weekly engagement, he didn't suspect a thing. Goodbye red player.

TUESDAY: Everyone tried to get in on the action today. The girl masquerading as the Dark Elf (who was wearing either black or red) and the sociology major lived to see another day. The purple player was able to catch the obliging yet naive green player by calling her and pretending he had car trouble.

WEDNESDAY: The biology major (who was still "alive") was surprised to hear that the Psychology major, who wasn't Anabel the art history major, ambushed Night Stalker as he slept in his dorm.

THURSDAY: The black player was declared the victor after luckily spotting "Billy" on his way to Mammalian Physiology, a class required by his major.

If you've never tried these before: 
has some  general hints for how to solve these type logic problems.

#5

Today's problem comes from AoPS:

The lockers in a school are numbered in order from 1 to 1000. Initially they a are all closed. There are 1000 students in the same school. The 1st student goes through the school and opens every locker. The 2nd student goes through the school and for every 2nd locker if the locker is closed she opens it and if it is opened she closes it. The 3rd student does the same for every 3rd locker, the 4th for every 4th locker and so until all 1000 students have gone through the school. After all the of the students have finished, how many lockers are open?


Hints: Try this for smaller numbers and see if you can find the pattern.

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