Reva Reverse

Ada looked up with her marker still poised, asking what that meant when they didn't know where the back door was.

"Sure you do," Reva said, shifting her weight until the wooden chair squeaked. "You don't know the first number, but you know the last one is twenty, so begin where you know something."

The four of them stared at the glowing screen, and Rue asked if you could really do a math problem backwards. She sounded deeply skeptical, as if she were waiting for some kind of trick.

"Almost every 'find the start' problem is really a 'walk back from the end' problem in disguise," Reva said. She took a loud bite of an apple. "You just have to learn how to undo what has already been done."

Reva spun her chair around to face the camera properly, pointing a green-sleeve-clad finger directly at the screen.

"Going forward, you are stuck before step one because the starting number is a mystery. But the end is sitting right there. It is twenty, so stand at the finish line and ask what had to happen right before you got there."

Ada looked down at the riddle, noting that the very last step was halving the result.

"Exactly," Reva said. "If they halved a number to get twenty, what was the number before that?"

"Forty," Sef said instantly, his chair legs hitting the floor with a loud thud because forty divided by two is twenty.

"Right," Reva said. "When you are working backwards, halving turns into doubling, so you do the opposite of each step in reverse."

She took another bite of her apple, explaining how she used to think you had to solve everything front-to-back like reading a book.

"But some problems are built backwards, hiding the beginning and handing you the prize at the end. Trying to force your way forward through them is exhausting, but starting at the finish lets the door open from the other side."

"So we just undo the operations?" Niko asked, his eyes wide as he looked at the board.

"Undo each step in reverse," Reva said. "They added eight, so you subtract eight. They doubled, so you halve. Opposite by opposite."

Ada wiped her whiteboard clean, leaving a dark smudge of ink across her palm as she wrote twenty on the right side.

"We start at twenty," Ada said. "The step right before was halving, so we undo it: double twenty is forty." She drew a backward arrow and wrote forty.

"Before that, they added eight," Rue said, "so we do the opposite and subtract eight to get thirty-two." She grabbed a blue marker and wrote thirty-two next to another backward arrow.

"And before that, they doubled," Niko said, practically vibrating as he grabbed the marker from Rue. "The opposite of doubling is halving, so half of thirty-two is sixteen." He wrote sixteen at the far left of the board.

The basement room went completely quiet, except for the low, steady hum of the laptop fan on the table.

"Sixteen," Sef whispered. "The starting number is sixteen, and we just walked in through the exit."

"Test it," Ada said, her heart beating faster. "Run it forward to make sure it actually works."

Niko took the marker and wrote: sixteen doubled is thirty-two, plus eight is forty, halved is twenty. He dropped the marker onto the table, staring at the clean chain of numbers.

"It works," Sef said. "It's actually right."

"It was right the whole time," Reva said, "just waiting for you to come at it from the other end."

Ada stared at the neat line of numbers, the confusion in her head replaced by a strange, cool quiet. It felt like solving a maze by tracing her finger from the exit back to the entrance, and it was almost too easy.

"Feel what just happened?" Reva asked, setting her apple core down on her desk. "Going forward, you couldn't take step one because the start was a blank wall, but you knew the end for certain. You stood in the one place you were sure of and walked back, one undo at a time, until the door opened."

"It felt so stuck before," Ada said, "and then it just flowed."

"That's the secret," Reva said. "It turns 'I have no idea where to begin' into 'I know exactly where I am going.' Knowing your destination, even when the path is foggy, is a completely different feeling than being lost, because you aren't guessing anymore. It is calm."

Rue reached out and traced the backward arrows, mentioning that her dad did something similar when he cooked dinner.

"He figures out when we want to eat, then works backward to decide when the chicken goes in the oven."