# Scrabble with prime numbers!

#### How to Play

Overall, gameplay is very similar to typical Scrabble; however, unlike typical Scrabble, you'll be using digits instead of letters (we'll cover your tile bag later). The objective is to achieve a higher score than your opponent while building prime numbers:

A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. For example, 5 is prime because the only ways of writing it as a product, 1 × 5 or 5 × 1, involve 5 itself.

For clarity, a number is considered valid, only if it is prime.

#### Scoring

Numbers are awarded a base score represented as the sum of their tiles. For example, take the number $$127$$, assuming that none of the digits occupy a multiplying tile, the base score for the number is $$1 + 2 + 7 = 10$$.

Additionally, per traditional Scrabble rules, if all 7 of a player's tiles are played in a single turn, an additional 50 points are awarded.

#### Multiplier Tiles

As with traditional Scrabble, there are multiplier tiles on the board that impact the base score:

• A double digit (DD) tile doubles the value of the digit occupying it.
• A double number (DN) tile doubles the entire base score of the number played.
• A triple digit (TD) tile triples the value of the digit occupying it.
• A triple number (TN) tile triples the entire base score of the number played.

If we calculate the total score for the number $$127$$, assuming the $$2$$ occupies one of these tiles per calculation, the scores would be $$12$$, $$20$$, $$14$$ and $$30$$ repsectfully.

#### Fibonnaci Primes

Fibonnaci primes are special in that they are both Fibonacci numbers, and prime numbers. When one is played, the resulting total score is doubled. Take the number $$131$$ for example; assuming that it doesn't occupy any multiplier tiles, the total score is $$1 + 3 + 1 = 5$$ and the final score is $$(1 + 3 + 1) \cdot 2 = 10$$.

#### The Tile Bag

The tile bag is distributed based on Benford's law and contains 110 tiles:

$$1$$x30, $$2$$x17, $$3$$x13, $$4$$x10, $$5$$x8, $$6$$x7, $$7$$x6, $$8$$x5, $$9$$x4, $$0$$x8 and 2 blanks.

#### End Game

As with traditional Scrabble, the game ends when:

• There are no valid moves remaining.
• A player runs out of tiles.
• A player concedes.

#### The Puzzle

Now, for the moment you've been waiting for! What is the highest achievable score in a single play? For your answer, you can assume that the board already contains a number that is advantageous to your goal with relation to your tile rack.

• This seems very difficult to prove optimal - do you have reason to believe there's a "nice" solution to this puzzle?
– Deusovi
Sep 28, 2021 at 0:13
• @Deusovi An optimal solution isn’t required; the challenge in finding the answer is what I found “nice” about it. Sep 28, 2021 at 0:21
• Alright, this version of the question looks good! It's clearly stated, and searching the state space of all possible turns is more reasonable than all possible games.
– Deusovi
Sep 28, 2021 at 2:04
• However, the tile bag distribution is not so good, because the distribution of the digits in prime numbers is close to uniform. In fact, Benford's law makes sense only for first digits of numbers. Additionally, primes in decimal notation can end with 1,3, 7 or 9 only. The scoring is not too fair either. Namely, the 0 digit tile should not definitely be worth 0 points. Sep 28, 2021 at 9:03
• Side note: 131 isn't actually a Fibonacci prime, n(131) ie. the 131st number in the Fibonacci sequence, is prime. The Wikipedia article is very confusing in that regard. Sep 28, 2021 at 14:31

EDIT2: My solution was already accepted, but (I assumed) far from optimal. I finally made myself a prime checking algorithm and checked the 32 numbers that allow maximum scoring for the main 'word'.

No less than 5 are prime! So one can score the maximum of 136*27 for the main word alone (with 2 9s and an 8 on the triples for maximum secondary scoring)

Edit: Score increased to 3532

Using the prime of TwoBitOperation, the score can be increased by placing in the 2L squares, and by getting multiple numbers counted.
Below (86+6) * 27 (main prime) + 32 * 3 (top prime) + 32 * 3(bottom prime) + 50 (7 tile bonus)= 2726
(Probably still far from optimal)
(green for blanks, other colored tiles are the played tiles)

Using the list of Left-truncatable primes from OEIS to get some big primes, and trying to maximize secondary scoring, I believe I have a valid score of 3532 (the green 7s are the blanks):
110 * 27 + 49 * 3 + 7 + 28 + 57 * 3 + 13 + 23 + 41*3 +50bonus

• Great answer! I calculated your score as $2534$ thanks to the doubling of two $3$ tiles and the 7 tile bonus of 50 points added at the end. While still higher than TwoBitOperation's answer, I'm not seeing how you arrived at $2676$, can you elaborate on that? Sep 28, 2021 at 12:53
• @Tacoタコスyou failed to notice that this solution also puts down two new horizontal primes at the top and bottom (9887). which is the 32 * 3+ 32 * 3 in his calculation. but he didn't count the 7 tile bonus so it's even higher
– Ivo
Sep 28, 2021 at 13:09
• @IvoBeckers great catch, thanks for pointing it out! Retudin, I recommend editing your answer to make that clear for future readers, but still a great answer! Sep 28, 2021 at 14:24

This is a huge question, and I think people can help each other here so I'm going to start with some initial strategy:

I'm attempting to build a 15 digit prime number with a high total digit sum. This prime will run along the edge of the board so that all 3 TN tiles can be hit at once. As such, this 15 digit prime would ideally have primes as substrings, because only 7 tiles can be played at once. There are no 15 digit Fibonacci numbers, and hitting all 3 TNs is more important than a 2x.

I have set up this board:

Note: One of the sixes in this set-up must be a blank.

Which allows for this move:

For a score of 2372 (Edit: confirmed by OP)

• It would also be nice if each tile on a TN square prepended one digit to a prime (played horizontally in your diagram) so as to make a longer prime. The score of each such number would be tripled (sextupled if also Fibonacci) and contribute to your total score. Sep 28, 2021 at 10:42
• Let me start by saying great answer! I'd like to point out a few things though. The first is that instead of using a blank to represent one of the sixes in your scoring prime, use it to represent a number already in play elsewhere (such as $367$); this adds $6$ to your base score. Additionally, the rule for placing all 7 tiles still applies, so you get an additional 50 points for that. Accounting for both your score would be $2534$. Sep 28, 2021 at 12:46
• Also did you use the palindromic primes as part of your initial strategy or did it come later? Sep 28, 2021 at 12:46
• Correction to my previous comment; the DD (2L) squares weren't in play, so the base score has to be reduced by six; this brings your score back down to $2372$. I didn't notice it at first lol Sep 28, 2021 at 12:49
• @Tacoタコス My comment meant that the blank 6 should be played in a move other than the scored move, sorry if that wasn't clear. As for palindromes... it wasn't part of the initial strategy, but it definitely made setting up the board easier. Sep 28, 2021 at 14:33