Stones on an infinite chessboard

We were inspired by this Numberphile video by Neil Sloane today.

Starting with 2 ones, what is the maximum number you can place on an infinite grid according to the rule:

“You can place a number if all the numbers immediately around it add up to itself”

Here is a way to get up to 4:

But then we are stuck as there is nowhere to put the 5. Can you do better than 4?

You can choose where to place the 2 ones to start with, for example:

or


This puzzle can be tried with any number of 1s to start with.

What is the maximum number you can place starting with three 1s?

If the maximum number you can reach is a(n) for n starting 1s, an interesting question is what is the best lower bound we can find for a(n).

Imagine this diagram continuing …

If we split it into sections we have demonstrated that a(n)>=3n-3

Can you find a higher lower bound for a(n) by drawing a different pattern that would continue indefinitely?

Random walk for Pi Day

Today we explored one of the amazing ways that pi crops up where you wouldn’t expect it.

We were inspired by this video to write a program in Trinket that simulated a random walk.

The rules are that you start at the origin and toss a coin. If you get a Head you go to the right by one, if you get a Tail you go to the left by one. If you did this for say 1000 steps, 1000 times, the average final position would be 0. But what would if we took the positive distance from the origin at each stage and averaged those values? It involves pi! Watch the video for more info.

Card games

We played a “perfect strategy” game today where everyone theoretically has an equal chance of winning.

The game is called 91 and the rules are below.

We also played a couple of “collaborative games” where all players work together to try and beat the cards. The two we played were German games called The Game and Hanabi.

The Game

Hanabi

Rules for 91:

Number of players: 2-7

Materials: 1 card deck for 2-3 players; 2 card decks for 4-7 players (no jokers).

Players and Cards: The game is for two to seven players, using a complete suit from a standard 52-card pack for each player plus one extra suit. Two decks are needed for four or more players. Cards rank Ace (low), 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K (high). As a prize, the Ace is worth 1 point, cards 2-10 face value, Jack 11, Queen 12 and King 13.

Setup: The cards are sorted into suits. One suit (traditionally diamonds) is shuffled and stacked face down as a prize pile. Each of the other players takes one complete suit.

Play: The top card of the prize pile is turned face up (so you see its value). Then each player selects a card from their hand with which to bid for it and places it face down (so you can’t see its value). When all players are ready, the bid cards are revealed simultaneously, and the highest bid wins the prize card. The bid cards are then discarded and the prize card is placed beside the player who won it. The next card of the prize pile is turned face up and players bid for it in the same way. If two players put the highest bit, the bid cards are discarded but the prize card remains on offer. A new prize card is turned face up and all players then make their next bid for the two prize cards together, then for three prize cards if there is another tie, and so on. If any of the players’ last bid cards are equal, the last prize card (and any others remaining from immediately preceding tied bids) are not won by any player.

Scoring: When all players run out of bid cards, the play ends. Each player totals the value of the diamonds they have won in bids (Ace=1, 2- 10 face value, J=11, Q=12, K=13) and the greater total wins the game.