It might not yet be the phenomenon that is Wordle, but for hundreds of thousands of players, Digit Party has scratched the itch of a casual brain teaser to break up their day.
Players arrange numbers on a 5-by-5 grid, earning points whenever identical numbers touch on adjacent or diagonally connected squares. They can then compare their score to the puzzle's maximum score that the game spits out at the end of a round.
There's just one problem: The game was lying. Or rather, its creators were. For more than three years, Vincent Vatter , Ph.D., at the University of Florida and Robert Brignall , Ph.D., at The Open University in the United Kingdom didn't know how to calculate the true high scores.
So they fudged it.
"We knew there was this issue, but we didn't know how to fix it," said Vatter. "We had a number we knew nobody could beat. Most of the time it was actually reachable, but about 5% of the time it wasn't, and we had no way to tell the difference. We knew it this whole time, and it bugged us."
Now, Vatter and Brignall report that they've discovered how to compute the true high score of every possible combination of numbers. Out of the last three years' 1096 daily puzzles, the game "lied" 55 times. The average error was just 2 points or so; the maximum was 6 points. With a game's typical maximum score of around 150 to 200 points, the wrong scores were off by a percentage point or two.
The team published their findings Aug. 25 in Math Horizons.
Although the stakes are small for this casual puzzle, the underlying mistake is common in optimization. The best solution for each individual part of a problem doesn't always combine into the best overall solution, a challenge that also appears in scheduling, logistics and manufacturing.
In Digit Party, the player scores based on how well they arrange the same number next to itself. Three 8s packed together score 8 three times, or 24. But an 8 by itself is worthless. So the problem to solve is how to efficiently pack all the same numbers together while fitting on a single 5-by-5 board.
Vatter likens it to packing a suitcase. First you figure out the best way to pack your shirts, then the ideal arrangement for your suits. Most of the time the shirts and suits pack together just fine.
"In the other 5% of cases, the suitcase is too crowded, and something has to get crumpled," Vatter said. "Then the question is what to crumple. Do you crumple the $10 t-shirt, or the suit? Our job was to find a way to compute the smartest thing to sacrifice for every packing list you could be handed."
The solution came from reducing the apparent complexity. The game can deal you any of 13.9 million different sets of digits. But rather than solving every possible board, they grouped boards that shared the same pattern of repeated numbers. Those groups reduced to just 1,291 underlying mathematical problems, and all but 400 required no trade offs at all.
For those remaining 400 cases, the researchers computed every worthwhile trade-off between competing arrangements, allowing the game to instantly identify the true maximum score. For example, if a player can't fit all the 4s and all the 8s together at once, it's usually better to drop a 4, because the 8 will be worth more at the end.
Finding the true maximum score turned out to be the easy problem. Discovering the best strategy for players — who only see one move ahead — remains unsolved.
"We still have no idea how to play," Vatter said. "Robert and I play quite differently, and nobody knows who's better."
