A problem inspired by Jay B

Number Theory Level 2

This is a problem inspired by Jay B. You can check out his similar problems!

There are 45 balls in a bowl. You can choose to take 23 or 5 balls in and out of the bowl, but you can only do one move at a time.

Only the 45 balls in the bowl can be used, and no other balls are used.

By following these rules, is it possible to have only 1 ball remain in the bowl?

Yes, more than 10 moves are needed Yes, less than 10 moves are needed It is not possible

1 solution

David Vreken
Jan 23, 2020

Yes, less than $10$ moves are needed:

$1) \text { } 45 - 23 = 22$

$2) \text { } 22 + 5 = 27$

$3) \text { } 27 - 23 = 4$

$4) \text { } 4 + 5 = 9$

$5) \text { } 9 + 5 = 14$

$6) \text { } 14 + 5 = 19$

$7) \text { } 19 + 5 = 24$

$8) \text { } 24 - 23 = 1$

Sir, can you please explain me how you reached this? I don't think this problem can be solved without trial and error!

Vinayak Srivastava - 1 year ago

There was a little bit of trial and error. Essentially I needed to solve $5x - 23y = 1 - 45$ , which rearranges to $23y - 5x = 44$ , a linear Diophantine equation where one solution is $x = 5$ and $y = 3$ . Since $x + y = 8 < 10$ , I knew there could be a solution, as long as I was careful to arrange the moves so that the running total of balls was not negative and did not exceed $45$ .

David Vreken - 1 year ago

A problem inspired by Jay B

1 solution

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