Euler's Officers

Full rules

1. Overview
2. The square
3. Setup
4. Your turn
5. Not a word
6. Dead ends and Euler's trap
7. End of the game
8. Difficulty levels
9. Tips for beginners
10. Variants

Overview

Euler's Officers is a cooperative puzzle for a team of two to four: you win or lose together. You use only the 16 officers of the deck – the four Aces, Kings, Queens and Jacks – and lay them out as a square of four rows and four columns. In the finished square, every row and every column must hold each rank once (one Ace, one King, one Queen, one Jack) and each suit once (one ♠, one ♥, one ♦, one ♣). On the Expert level the two diagonals must do the same.

The cards are shared out among you, and you take turns placing them one at a time – without a word. It sounds easy, but a square can become impossible long before anyone notices. A few coins let you take back cards that turn out to be in the wrong place.

• Players: 2–4, best with 2 (1 in the Solo variant).
• You need: the 16 Aces, Kings, Queens and Jacks from a standard deck, up to 3 coins as take-back tokens, and a table with room for a 4 × 4 square of cards.

Euler's Officers is a new game written for this site and not yet playtested. It turns a card puzzle that was printed as early as 1725 – and later made famous by Leonhard Euler as the problem of the 36 officers – into a game for a silent team. The number of take-back tokens is the first thing to tune (see "Difficulty levels").

The square

• A line is a row (across) or a column (down). On the Expert level, the two diagonals from corner to corner are lines too.
• The rule of the square: no line may ever hold two cards of the same rank or two cards of the same suit. Since each line has four places and there are four ranks and four suits, a full line holds every rank and every suit exactly once.

One finished square (it doesn't meet the diagonal rule – all four Aces lie on the diagonal from top left to bottom right – so it doesn't count on Expert):

There are 6912 different finished squares without the diagonals and 1152 with them, so there is plenty of room – and plenty of room for mistakes.

Setup

1. Take the 16 Aces, Kings, Queens and Jacks out of the deck. The other cards are not needed.
2. Choose a level and put its take-back tokens in the middle: on Normal 1 coin (see "Difficulty levels").
3. Shuffle the 16 cards and deal all of them, one at a time, face down: with 2 players 8 each, with 3 players 6, 5 and 5 (the first player gets 6), with 4 players 4 each. This is your hand: look at it, but don't show it to the others.
4. The player left of the dealer begins – in these rules that is Anna. Then play goes clockwise.

The square starts empty; just leave room for four rows of four cards.

Your turn

On your turn you do one of two things:

• Place a card: put one card from your hand face up on any empty place of the square – the rule of the square must still hold. The first card of the game may go anywhere. You may leave gaps and fill them later.
• Take back: return one take-back token to the box, pick up any one card from the square and add it to your hand. That is your whole turn.

Then it is the next player's turn. A placed card stays where it is until someone takes it back. If your hand is empty, you are skipped.

• Stuck: if none of your cards may go on any empty place, you must take back. If no token is left, the team has lost.
• Slips: if someone places a card that breaks the rule of the square, they take it back into their hand at once and place another card instead. That costs nothing.

Not a word

During the game you may not say anything about your cards or your plans: no "I've got the Ace of hearts", no "leave that place for me", no pointing, no gestures and no groaning when someone places a card. You may talk about the rules and about whose turn it is, and you say "take back" when you take a card back. Except on Expert, take as long as you like to look at the square.

You know more than it seems: with 2 players, your partner holds exactly the cards that are neither in your hand nor in the square. With 3 or 4 players you know which cards the others hold together, but not who holds which. Where someone puts a card tells you which square they have in mind.

Dead ends and Euler's trap

A square can become impossible to finish long before anyone is stuck. Three checks catch most dead ends:

1. Every empty place can still take at least one of the cards not yet in the square.
2. Every card not yet in the square still has at least one empty place where it may go.
3. Every line can still get each rank and each suit it is missing.

End of the game

• You win together as soon as all 16 cards are in the square.
• You lose when a player is stuck and no take-back token is left. If you all agree that the square is dead and you have no token left to repair it, you may say so and stop – that counts as a loss.

Every deal can be won. However the cards are shared out, as long as the square can still be finished, every card still in play has at least one safe place – the place it has in a square that finishes this one. So you can never be forced to lose by the deal, only by a card in the wrong place. A take-back repairs one such card.

Difficulty levels

On Expert, keep time with a phone or a sand timer: whoever has neither placed nor taken back after 30 seconds loses the game for the team.

The budget: you need 16 turns to place 16 cards. Each take-back costs one turn and puts one card back into a hand, so it costs the team two turns in all – but it can save a square that has gone wrong. With the diagonals only 1152 of the 6912 squares are left; on the other hand, the diagonals often show a mistake earlier.

In our computer simulation – 2500 deals per level and number of players – the team was a simple bot: before every card it runs the four checks above, then places a random card on a random place that passes them, and when its checks show a dead end, it takes back. With two players it won 80% of its games on Easy (without talking), 57% on Normal and 41% on Hard; with three or four players it did a few points better (up to 85% on Easy). On Expert it won 43% – without the clock, which a program doesn't feel. The diagonals hardly changed its chances: they rule out many squares, but they also show mistakes sooner. The bot's weakness is that it doesn't plan: a bot with the same checks that always placed its card next to cards already in the square won 90% of its games without a single take-back (83% with diagonals). So once you have the knack, move up a level – and if even Expert is too easy, shorten the clock.

Tips for beginners

• Try it without the spoiler first. The folded fourth check above is the secret of the puzzle – open it when the three checks no longer save you.
• Build from a corner. Filling the square line by line keeps the picture clear for everyone; scattered cards hide dead ends.
• Read your partner. With two players you know their hand exactly – ask yourself where their cards can still go, and don't take the only place one of them has left.
• Fill forced places first. If a place can take only one remaining card and you hold it, place it there.
• With diagonals, every finished square has one more property: each quarter of the square (the four 2 × 2 corner blocks) also holds every rank and every suit. Use it as a fifth check.

Variants

All optional – agree before you start.

• Keep-free coin (easier): each player has a coin of their own. At the end of your turn you may put it on an empty place or move it – it means "please leave this place for me". It is the only signal you may give.
• Open hands (easier): play with your hands face up, but still without talking.
• Solo: one player holds all 16 cards and places them one after another – the puzzle as Ozanam printed it.
• Ozanam's riddles (puzzle starts): before dealing, lay out four cards as a start, then deal the other twelve. Each of these starts can be finished in exactly one way. Without diagonals: A♣ in row 1, column 1; J♠ in row 2, column 2; A♥ in row 3, column 2; Q♦ in row 3, column 4. With diagonals: A♦ and J♣ in row 1, columns 1 and 2; J♠ in row 3, column 1; K♣ in row 4, column 1. Starting cards may not be taken back.