Ursula's Spoiler[]
Computing the reduced row echelon form of the result matrix results in one dependent column and two zero vectors:
| B | E | T | R | A | Y | A | L | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| BB | T | Y | LLL | 1 | ||||||||||
| EE | TTT | RRR | YY | AAA | L | 1 | 1 | |||||||
| BBB | EEE | R | Y | 1 | 3 | |||||||||
| B | T | YY | LLL | 1 | ||||||||||
| E | TTT | RRR | YY | L | 1 | 1 | ||||||||
| B | TT | Y | LLL | 1 | ||||||||||
| B | RRR | A | YYY | |||||||||||
| E | AAA | Y | AAA |
Using this, we can determine that there are 16 possible solutions (that don't involve clicking the same letter more than 3 times) from the initial state [3,1,3,1,1,3,1]. Here they are listed below, in order of fewest clicks (7) to most (15):
| Clicks | B | E | T | R | A | Y | A | L |
|---|---|---|---|---|---|---|---|---|
| 7 | BB | E | Y | AA | L | |||
| 7 | BB | A | AAA | L | ||||
| 7 | B | EEE | R | A | L | |||
| 7 | B | R | Y | AAA | L | |||
| 7 | EEE | RR | Y | L | ||||
| 7 | EE | RR | A | A | L | |||
| 11 | BBB | EE | RRR | Y | A | L | ||
| 11 | BBB | E | RRR | A | AA | L | ||
| 11 | BB | EEE | AA | YYY | L | |||
| 11 | B | EE | R | AA | YYY | A | L | |
| 11 | B | E | R | AAA | YY | AA | L | |
| 11 | E | RR | AA | YYY | AA | L | ||
| 11 | RR | AAA | YY | AAA | L | |||
| 15 | BBB | EEE | RRR | AAA | YY | L | ||
| 15 | BBB | RRR | AA | YYY | AAA | L |
Of interesting note is that the "T" letter should never be pressed.