Author: Emily Fleming

Christmas trees solution

There must be 4 decorated trees and 5 not decorated.

The first assumption that can be made is that either Penny, Olivia or Noah are lying, because if Penny is lying, everyone else is truthful, and if Matt is lying, Penny states that Noah or Olivia are also lying, so either one of those three are lying, which means there are too many liars.

Therefore, we have two truths: there is one more decorated tree than undecorated, and the number of trees is prime, what Matt says. Starting with all primes smaller than ten, there could be 2 undecorated and 3 decorated trees, 4 undecorated and 5 decorated trees, or 6 undecorated and 7 decorated trees.

If Olivia is telling the truth, the total is 9 (4+5) , which also means that Noah is telling the truth ( 8 < 9 < 12), and Penny is lying. In the other cases, both Noah and Olivia must be lying at once(2+3 < 8 and 6+7 > 12, none of these possibilites contain 4).

The only possible solution is when Penny is lying, and we’ve seen that in this scenario, there are 5 decorated trees and 4 undecorated trees.

Snowman’s football match (solution)

The possibilities are:

(Wins, draws, goals)

Rudolphs: (0, 0, 8) or (0, 1, 3)

Comets: (0, 0, 14) or (0, 1, 9) or (0, 2, 4) or (1, 0, 4)

Vixens: (0, 0, 9) or (0, 1, 4)

 

If there were 3 games, then there would be 30 points

for wins and draws, leaving just 1 point for goals,

so that is not possible, since each team scored in each game.

 

Thus, there were only 2 games,

yielding 20 points for wins and draws, 11 for goals,

which reduces the possibilities to

Rudolphs: (0, 1, 3)

Comets: (0, 2, 4) or (1, 0, 4)

Vixens: (0, 1, 4)

 

Rudolphs had a draw (and maybe a loss).

Comets had 2 draws or a win and no other match.

Vixens had a draw (and maybe a loss)

 

If Comets had 2 draws, then both games were draws,

but that is not possible since total goals was 11, which is odd.

So Comets had a win against one of the others,

and the other game was a draw between the Rudolphs and the Vixens.

 

The score of Rudolphs-Vixens draw then was either 1-1, 2-2, or 3-3

Since Vixens scored more goals than Rudolphs, they

are the ones who lost to Comets, and Rudolphs, only playing once,

scored all 3 goals in that game, equalling Vixens.

Comets scored all 4 of their goals in their only game, where

Vixens scored one additional goal.

 

The results were:

Comets  4, Vixens 1

Rudolphs 3, Vixens 3

Snowman’s football match

Snowman had never fully understood the points  system in football, and felt that the scoring of goals should be encouraged. His idea is that 10 points should be awarded for a win, 5 points for a draw and 1 point for each goal scored, whatever the result of the match.

This was tried with three teams: Rudolph, Comet and Vixen.

Each team scored at least one goal in every match and no team played another more than once.

Rudolph scored 8 points, Comet 14 points and Vixen scored 9 points.

Find the score in each match.

Solution is here.

Galton’s Board

galton board

This week we looked at an online simulation of the Galton board (or Bean Machine), which is a device where beads are dropped from a funnel at the top through ranks of nails.
Each time a bead strikes a nail it has a 50% chance to veer left and a 50% chance to veer right.

Each bead eventually drops in one of the column A, B, C, D or E.

1. Do you think the probabilities for a bead to land in A,B,C,D or E are equal ?
2. If not which column has the highest probability ?
3. Can you do a computer simulation of 10,000 beads dropping in a Galton
board using scratch or Python ?

Dice simulation

When you throw two dice and add the numbers together what number are you most likely to get? What is the smallest number you can get? What is the biggest? Are all numbers equally likely?

We used python to simulate rolling a dice 100 times and plotted the results using pygal. Step by step instructions are here.

A big challenge would be to program the following game on python:

Pig Game

The game of Pig is a two player game played with two six-sided dice. The object of the game is to reach 100 points of more. Play is taken in turns. On each person’s turn that person has the option of either:

  1. Rolling the dice: where a roll of two to six is added to their score for that turn and the player is given the same choice again; or a roll of 1 loses the player’s total points for that turn and their turn finishes with play passing to the nexxt player.
  2. Holding: the player’s score for that round is added to their total and becomes safe from the effects of throwing a 1. The player’s turn finishes with play passing to the next player.

 

Medal Muddle

medal

We really enjoyed working on the following puzzle from Nrich.

Thirteen nations competed in a sports tournament. Unfortunately, we do not have the final medal table, but we have the following pieces of information:

1. Turkey and Mexico both finished above Italy and New Zealand.

2. Portugal finished above Venezuela, Mexico, Spain and Romania.

3. Romania finished below Algeria, Greece, Spain and Serbia.

4. Serbia finished above Turkey and Portugal, both of whom finished below Algeria and Russia.

5. Russia finished above France and Algeria.

6. Algeria finished below France but above Serbia and Spain.

7. Italy finished below Greece and Venezuela, but above New Zealand.

8. Venezuela finished above New Zealand but below Greece.

9. Greece finished below Turkey, who came below France.

10. Portugal finished below Greece and France.

11. France finished above Serbia, who came above Mexico.

12. Venezuela finished below Mexico, and New Zealand came above Spain.

We came up with different strategies to sort out the medal table, and we were largely successful eventually, but we were all impressed by a quick way to solve it!