# The Mathematical Ninja and the twenty-sixths

The Mathematical Ninja played an implausible trick shot, not only removing himself from a cleverly-plotted snooker, but potting a red his student had presumed safe and setting himself up on the black. Again. “One!” he said, brightly, and put some chalk on the end of his cue.

The student sighed. “You were going to tell me about how to work out twenty-sixths, sensei.”

“Eight!” The white smashed into the pack, and the remaining fourteen reds wound up conveniently in line with pockets. “Ah, yes, twenty-sixths. You know about thirteenths, of course?”

“Yes, sensei, you multiply by 77, knock one off, and stick the complement to 999 on the end.”

“Nine!”

“Pretty sure it was 999, sensei. Oh, you meant the score.”

“Sixteen!”

“So, let’s say I had to work out …”

“Seventeen!”

“OK, seventeen twenty-sixths. That’s midway between… 8 and 9 thirteenths?”

“Twenty-four. You could do it that way.”

“How would you do it?”

“Twenty-five. I’d think of it as $\frac{13}{26} + \frac{4}{26}$.”

“Oh! I see where you’re coming from.”

“Thirty-two.”

“So it’d be two thirteenths – which is $0.\dot 153 84\dot 6$, add a half, making $0.6 \dot 538 46\dot 1$, I suppose.”

“Thirty-three.” The Mathematical Ninja was now swerving the cue ball into several cushions before entertaining the idea of potting reds. “That’s right.”

“How about for the low ones, like $\frac{3}{26}$?”

“Forty,” said the Mathematical Ninja, sternly. “Think about it for a moment.”

“Aha! That would be $\frac{13}{26} - \frac{10}{26}$.”

“Forty-one. Possible, but probably more work than you want to do.”

“Or… how about $\frac{16}{26} - \frac{13}{26}$?”

“Forty-eight. That’s more like it.”

“Are you allowed to make the balls jump off the table like that?”

The Mathematical Ninja shrugged. “I just did.”

Making them juggle seemed to border on the unfair, thought the student. “So it’d be eight-thirteenths (which is a tricky one… 5-11-6 is 616 so $0.\dot615 38\dot4$) minus a half, which is $0.1\dot 15 384 \dot6$.”

“Forty-nine. Yep, that sounds about right.”

“You’re going to clear up from here, aren’t you?”

“Fifty-six. It would go a lot quicker if you didn’t keep interrupting with your twenty-sixths.”

“Sorry, sensei.”

“Fifty-seven.”