Logic
(1 min 30 secs) The day before two days after the day before tomorrow is Saturday. What day is it today?
Friday
day before tomorrow = today, so it becomes
"the day before two days after today"
which is the same as "the day after today"
which is the same as "tomorrow"
so tomorrow is Saturday, meaning today is Friday.
(30 secs) What is the definition of "prime" and "composite"?
Prime: A number that is only divisible by 1 and itself.
Composite: A number that is divisible by a number that's not 1 and not itself.
(45 secs) 1. What are the two things that make up a "graph"?
2. What is the purpose of graphs?
1. Nodes/vertices and edges
2. To show the relationship between things
(30 secs) What does it mean for a number to be "divisible" by another number?
no remainder left / remainder = 0
TEAM VS TEAM
Your team will need to verify that they're not a robot. Your teammates will be doing a relay, where each teammate answers a question, then it moves on to the next person. Each team will have 5 minutes to answer as many questions as they can; whoever has the most questions answered by the end, wins! [if your teammate is stuck on a question, you can tag yourself in to help by doing 10 jumping jacks]
https://neal.fun/not-a-robot/
(2 mins) How many three-digit integers are exactly 17 more than a two-digit integer?
17
the numbers 100 to 99+17 = 116 (i.e. 100, 101, 102,..., 116) so in total, there are 17 numbers.
(30 secs) What is our current number system called?
"decimal" or "base-10"
(45 secs) In a fair coin flip, what's the probability of getting heads 6 times in a row?
1 / (2^6) = 1 / 64
(1 minute 30 secs) Is 1234567895 divisible by 11?
yes
TEAM VS TEAM
Send someone from your team to the front to get 50,000 clicks in a game within 5 minutes (there's a twist though!)
https://neal.fun/stimulation-clicker/
(3 mins) Five people were eating apples, A finished before B, but behind C. D finished before E, but behind B. What was the finishing order?
CABDE. Putting the first three in order, A finished in front of B but behind C, so CAB. Then, we know D finished before B, so CABD. We know E finished after D, so CABDE.
(1 min) What does the pigeonhole principle state?
If you have n+1 pigeons that you try to fit into n boxes (or holes), then one box must have at least two pigeons.
(1 minute 30 secs) Suppose you have 6 different people who form a line together. How many different arrangements are there?
6! = 6 x 5 x 4 x 3 x 2 x 1
(1 minute 30 secs) If you want to find out if a number is divisible by 66, what rules could you give? (Break your rules into smallest possible cases whenever possible)
Divisible by 2, 3, and 11
(Must state the rules for divisibility of 2, 3, and 11 too)
TEAM VS TEAM
I'll pick 5 people from each team, randomly. All of you will try to make the perfect circle. The team with the highest average wins.
https://neal.fun/perfect-circle/
(4 mins) You have twelve coins. You know that one (and only one) is a forgery, and is of a different weight to the other eleven coins. You have a balance allowing you to determine which of two sets of coins is heavier (but no other information).
first weight 6 coins and 6 coins
then from the first weighing, you know that one of the groups is heavier so pick that group and split it into 3 coins and 3 coins
then from the group of 3 coins that's heavier, split it into 1 coin and 1 coin in the next weighing.
(1 min) What does the fundamental theorem of arithmetic state?
Every integer (greater than 1) can be expressed as a product of prime numbers.
(3 mins) In how many different ways can 6 people be seated at a round table? (all people must be seated at once)
6! / 6 = (6 x 5!) / 6 = 5! = 5 x 4 x 3 x 2 x 1
(4 mins) How many factors does the number 360 have? (and what are they)
24 factors
1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360
TEAM vs TEAM
Your team will have 5 minutes to create the funniest word "starting from scratch" Whoever has the funniest word by the end wins.
https://neal.fun/infinite-craft/
(4 mins) You are in a room with two doors.
One door leads to freedom, the other to certain death. Guarding the doors are two guards:
- One guard always tells the truth
- One guard always lies You don't know which guard is which, and you don't know which door leads where.
You can ask ONE question to ONE guard to determine which door leads to freedom. What question should you ask?
'If I asked the other guard which door leads to freedom, what would they say?' Then choose the opposite door.
(3 mins) For the formula for the sum of positive integers from 1 to n (i.e. 1 + 2 + ... + n = [n x (n+1)] / 2, I showed you a proof in class.
How does the proof work, and which mathematician came up with it?
proof: create a copy of the sum, add everything together, then divide by 2.
mathematician: gauss
(4 mins) Suppose you have 5 different people who form a line together. In this group of 5 people, there's a mom and a son who must stay together at all times. How many different arrangements are there?
2 * 4! = 2 * 4 * 3 * 2 * 1 = 48
(3 mins) If a number, a, divides a number, bn, where n is a positive integer exponent (i.e. bn = b x b x ... x b [n times]) then does does a divide b?Explain your answer.
yes, and it's called euclid's lemma!
TEAM vs TEAM
You'll be spending Bill Gates' money. Whichever team has the most items after 5 minutes, wins.
https://neal.fun/spend/