An Additive Game

A deck of poker cards.

Here’s a game four people can play using a deck of poker cards.

• Split the deck by suit and give one to each player.
• Have every player return their Queen card.
• Let the game begin!

The objective of this game is to arrange your suit along the edges of a 4x4 square where every edge sums up to 23. For the purposes of our game we’ll take the Ace to have a value of one, the Jack to have a value of eleven and the King to have a value of thirteen.

Getting Started

An initial approach would be to first lay out the cards randomly and inspect them to see whether they match the description. That would look something like this:

A possible (though incorrect) arrangement.

Now to calculate our sums:

• Top edge: 2+10+13+11=36
• Left edge: 2+8+6+5=21
• Bottom edge: 5+3+1+7=16
• Right edge: 11+9+4+7=31

Unfortunately, none of our edges sum up to exactly 23. But we can try to swap cards between edges to get a correct arrangement. With a bit of luck this works out for us after performing the following swaps:

• Swap K and A
• Swap 9 and 10
• Swap 10 and 8
• Swap K and 8
• Swap J and A
• Swap 3 and 7
• Swap 4 and 6
• Swap 5 and 7

We end up with an arrangement that looks like this:

A solution to the Add23 Puzzle.

At each step I try to make a single edge get closer to 23 by swapping out its cards with those from another edge.

Taking it Further

Now, we could be satisfied with our solution and leave it at that. But where’s the fun in that! Are there other solutions? Is it possible to generate all of them? Answering these questions would require a different approach to the puzzle.

We’ll get to this in the Part II and do a deeper analysis of the puzzle. For now you can play around with the game here: https://docs.google.com/document/d/1o-gBtOrIidS1VcZkbYvzpo1zIJObY7O0cO3vXx-3K2M/edit?usp=sharing .

Read Part II here: https://newtonmigosi.medium.com/an-additive-game-part-ii-2a6a590c3a13