Scrabble 46

Here’s a puzzle from Matt Parker:

How many ways, from the 100 standard scrabble tiles, can you choose seven which total 46 points?’ Clarification: for this we’re asking you how many distinct scrabble hands (groups of 7 letters) there are that total exactly 46. So order does not matter and identical letters are indistinguishable.

The distribution of scrabble tiles and points is below. (Eg the letter A is worth 1 point, and there are 9 letter A tiles)

0 points – blank (x2)
1 point – A (x9), E (x12), I (x9), O (x8), U (x4), L (x4), N (x6), S (x4), T (x6), R (x6)
2 points -D (x4), G (x3)
3 points -B (x2), C (x2), M (x2), P (x2)
4 points -F (x2), H (x2), V (x2), W (x2), Y (x2)
5 points -K (x1)
8 points – J (x1), X (x1)
10 points -Q (x1), Z (x1)

Notice that if you have seven tiles adding to 46 then the average tile value is 46/7 which is 6 point something. So you must have at least one of the “big four” tiles: Q, Z, J, and X.

In fact you have to have all four. If you have only three (or fewer) then your highest three tiles can be worth no more than 28 points, so you need 18 points from your other four tiles. But the highest total you can get with four non big four tiles is 5+4+4+4 = 17.

So you need the big four, giving 36 points, and then you need three more tiles totaling 10 points. These can be:

  • 5 + 4 + 1 points — There is 1 way to get 5 points, 5 ways to get 4 points, and 10 ways to get 1 point, so the number of possibilities is 1 times 5 times 10 = 50.
  • 5 + 3 + 2 points — 1 way to get 5 points, 4 ways to get 3 points, 2 ways to get 2 points, number of possibilities is 1 times 4 times 2 = 8.
  • 4 + 4 + 2 points — 5 ways to get the first 4 points, but that leaves only 4 ways to get the next 4, and that gives you each pair of 4-point letters twice, and then there are 2 ways to get 2 points, so in total the number of possibilities is (5 times 4)/2 times 2 = 20.
  • 4 + 3 + 3 points — 5 times (4 times 3)/2 = 30 possibilities.

And that’s all. When I first thought about this puzzle I came up with the above and then realized I’d forgotten about the blanks, but then figured out the blanks can’t be used. If you have the big four plus a blank you need to get 10 points with two of the remaining tiles, and that can’t be done.

So the answer is 50 + 8 + 20 + 30 = 108.

Here they are:

Q Z J X K F A
Q Z J X K F E
Q Z J X K F I
Q Z J X K F L
Q Z J X K F N
Q Z J X K F O
Q Z J X K F R
Q Z J X K F S
Q Z J X K F T
Q Z J X K F U
Q Z J X K H A
Q Z J X K H E
Q Z J X K H I
Q Z J X K H L
Q Z J X K H N
Q Z J X K H O
Q Z J X K H R
Q Z J X K H S
Q Z J X K H T
Q Z J X K H U
Q Z J X K V A
Q Z J X K V E
Q Z J X K V I
Q Z J X K V L
Q Z J X K V N
Q Z J X K V O
Q Z J X K V R
Q Z J X K V S
Q Z J X K V T
Q Z J X K V U
Q Z J X K W A
Q Z J X K W E
Q Z J X K W I
Q Z J X K W L
Q Z J X K W N
Q Z J X K W O
Q Z J X K W R
Q Z J X K W S
Q Z J X K W T
Q Z J X K W U
Q Z J X K Y A
Q Z J X K Y E
Q Z J X K Y I
Q Z J X K Y L
Q Z J X K Y N
Q Z J X K Y O
Q Z J X K Y R
Q Z J X K Y S
Q Z J X K Y T
Q Z J X K Y U
Q Z J X K B D
Q Z J X K B G
Q Z J X K C D
Q Z J X K C G
Q Z J X K M D
Q Z J X K M G
Q Z J X K P D
Q Z J X K P G
Q Z J X F H D
Q Z J X F H G
Q Z J X F V D
Q Z J X F V G
Q Z J X F W D
Q Z J X F W G
Q Z J X F Y D
Q Z J X F Y G
Q Z J X H V D
Q Z J X H V G
Q Z J X H W D
Q Z J X H W G
Q Z J X H Y D
Q Z J X H Y G
Q Z J X V W D
Q Z J X V W G
Q Z J X V Y D
Q Z J X V Y G
Q Z J X W Y D
Q Z J X W Y G
Q Z J X F B C
Q Z J X F B M
Q Z J X F B P
Q Z J X F C M
Q Z J X F C P
Q Z J X F M P
Q Z J X H B C
Q Z J X H B M
Q Z J X H B P
Q Z J X H C M
Q Z J X H C P
Q Z J X H M P
Q Z J X V B C
Q Z J X V B M
Q Z J X V B P
Q Z J X V C M
Q Z J X V C P
Q Z J X V M P
Q Z J X W B C
Q Z J X W B M
Q Z J X W B P
Q Z J X W C M
Q Z J X W C P
Q Z J X W M P
Q Z J X Y B C
Q Z J X Y B M
Q Z J X Y B P
Q Z J X Y C M
Q Z J X Y C P
Q Z J X Y M P

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