Puzzle Tumblr

"Mathematics knows no races or geographic boundaries; for mathematics, the cultural world is one country." - David Hilbert

[jdgmiles]

The 50

#1 - Hey I Can Be the Answer #2 - Ambiguous Clock #3 - (a + a⁻¹) = 4 ⇒ (a⁴ + a⁻⁴) = ? #4 - Squares on a Chessboard #5 - Cubes on a Cheeseboard #6 - Bears on a Surfboard #7 - Twenty-digit Square with Eleven Leading 1s? #8 - Sum of Coefficients #9 - Vampire Hunter #10 - Chip Cooking Cunning #11 - Tetrahedron Edge Sums #12 - Up/Side/Down Journey #13 - Ten Easy Pieces #14 - Apple and Blackcurrant #15 - Blackcurrant and Apple #16 - Choosing Cards #17 - Smallest Number Using All Vowels #18 - Keyboard Word #19 - Bank Withdrawal #20 - Tie, Try, Snore #21 - Dicing With The Cube #22 - Printing With Dice #23 - Hanging a Print #24 - Seven Sequential Squares Summed #25 - Mastery of the Unseen #26 - Cycle of Arrows #27 - Build a Board to Beat Me #28 - Aᴮ - Bᴬ  = A + B #29 - A Table Divided #30 - A Table Saved #31 - Poisoned Wine #32 - “Skate where the puck’s going, not where it’s been” #33 - Broken Calculator #34 - Slot Machine #35 - Chopping a Triangle #36 - Tricolor Integers #37 - Figure Out the Function #38 - “Coin-Tossing” Game #39 - For Each Positive Integer k, Find the Smallest n Such That 2ᵏ|(5ⁿ-1) #40 - β ∈ ℝ such that β + β⁻¹ ∈ ℤ. Prove that βⁿ + β⁻ⁿ ∈ ℤ ∀ n ∈ ℕ #41 - Predict the Colour Card #42 - Dots and Plates #43 - Coins on a Table #44 - Last One Standing #45 - Robot Uprising #46 - Forcing Checkmate on an Infinite Chessboard #47 - Two Handshake Problems #48 - Cut the Cake #49 - Witch Hunt on a Tetrahedron #50 - Enclose the Most Valuable Region

#XX - “Mission Accomplished”

[jdgmiles]

Mission Accomplished

I decided that 50 was a nice number to end the blog on.

What started back in Uni with my “Wednesday Maths Problems” series has blossomed and I’ve learned a huge amount along the way. It’s been fun.

I’ll always be interested in this stuff - and continue collecting and sharing puzzles socially - but, for Puzzle Tumblr, I like 50 as the number to go out on and leave the site as a polished, finished entity.

If you’ve been through all 50 and want more, then there are plenty of places online, such as the excellent Math Riddles Subreddit. Or, hey, you can even get in touch with me and I’ll tell you whatever puzzle I’ve heard recently!

In parallel with posting the puzzles here, I had been maintaining a sister blog Solution Tumblr keeping up with the solutions.

However, there’s one final twist.

I haven’t posted the final five solutions there.

I figured that would give something for people to keep thinking about.

(If you’re desperate, you can email me and I’ll be happy to provide the solutions.)

All the best,

James Miles, Cambridge UK, August 2015

[jdgmiles]

#50 Enclose the Most Valuable Region

A wealthy sultan promises you all the land you can enclose in a 100km fence, beginning and ending at his palace.

Land more than 20km north of the palace is rich in oil and worth $2m per km2.

All other land in the kingdom is worth $1m per km2.

What shape maximises the value of the sultan's present to you?

[jdgmiles]

#49 Witch Hunt on a Tetrahedron

Three assassins are positioned on the edges of a regular tetrahedron.

The assassins are trying to find a witch, who is also on an edge of the tetrahedron.

They may all move along these edges, including through vertices to transfer from one edge to another.

The assassins know each other's locations, but not the witch's.

The witch knows the assassins' locations all times (and where they are about to move), but can only move P% as fast as them herself (0<P<100).

Is there a strategy that guarantees the assassins will find the witch in finite time? Does it depend on P?

[jdgmiles]

#48 Cut the Cake

I have baked a 30cm x 30cm square cake for my nine friends. It is 8cm tall.

Part 1: How can I cut the cake into nine pieces so that each friend gets an equal amount of cake?

Part 2: After icing the top and the sides but not the bottom, how can I cut the cake into nine pieces so that each friend gets an equal amount of cake and icing?

[jdgmiles]

#47 Two Handshake Problems

N people have a party.

1. On arriving everyone shakes hands with everyone else exactly once. How many handshakes are there in total?

2. On leaving, only some people shake hands. Prove that there are two people at the party who shake hands with the same number of people.

[jdgmiles]

#46 Forcing Checkmate on an Infinite Chessboard

You place pieces anywhere you like on an infinite Chessboard, then I place a King anywhere I like.

Using only Queens, how many do you need to be able to force checkmate?

Rooks?

Bishops?

Knights?

What is the minimum number of moves, for each?

[jdgmiles]

#45 Robot Uprising

121 computers arranged in an 11 x 11 grid have gone rogue!

Fortunately, any individual computer can be cured and made "friendly" again.

Furthermore, if a rogue computer is neighbouring two friendly computers then it will be persuaded to become friendly too. ("Neighbouring" = "sharing an edge".)

How many computers must individually be cured to return the entire network friendly again?

Can you prove a lower bound?

(Click the picture to play.)

[jdgmiles]

#44 Last One Standing

10 armed maniacs convene in a room.

At the word "go" they all simultaneously shoot a random other person.

This is repeated with the survivors until either everyone is dead or only one remains.

What is the probability of each end state?

[jdgmiles]

#43 Coins on a Table

There are 100 identical coins on a rectangular table, not overlapping, and such that it is impossible to add another coin without that coin overlapping another somewhere on the table.

Prove that the table can be completely covered (no gaps) with 400 such coins which may overlap.

[jdgmiles]

#42 Dots and Plates

I have 10 identical plates.

Can you place 10 dots on a table such that I cannot cover your dots with my plates without any overlapping?

[jdgmiles]

#41 Predict the Colour Card

Shuffle a deck of playing cards.

Guess if the top card is red or black, then turn it over. If you're right, you win a dollar. If not, you pay a dollar. Repeat for the rest of the deck, one card at a time.

What would be your strategy, and what is its expected return?

[jdgmiles]

#40 β ∈ ℝ such that β + β⁻¹ ∈ ℤ. Prove that βⁿ + β⁻ⁿ ∈ ℤ ∀ n ∈ ℕ

As title.

[jdgmiles]

#39 For Each Positive Integer k, Find the Smallest n Such That 2ᵏ|(5ⁿ-1)

Well go on, get to it.

[jdgmiles]

#38 “Coin-Tossing” Game

Charlie wants to buy a car for 2C dollars, but he only has x < 2C dollars. He decides to play a gambling game in the hopes of raising his money to 2C. The rules are as follows:

Charlie chooses how much to pay, then n fair dice are thrown. If the sum is even then he gets double his money back, if it is odd he gets nothing. He can play as many times as he likes.

Charlie bets all his money if he has less than C, otherwise he bets enough to make his total up to exactly 2C if he wins.

Determine the probability that Charlie reaches 2C before going bankrupt, as a function of x.

[jdgmiles]

#37 Figure Out the Function

Find a function f(x,y) : (ℝ,ℝ) → ℕ such that:

f(x,y) = f(y,z) implies x = y = z.

[jdgmiles]

#36 Tricolor Integers

Let's paint {1, 2, 3, ..., N} in red, blue and white, and so that each has more than a quarter of the set in its colour.

Can they be painted such that all x + y = z in the set has at least two of the same colours?