New Year's Countdown Day 25: Merry Christmas!

Number Theory Level 2

A well-known mathematical joke goes like this:

Q: Why do mathematicians always get Halloween and Christmas mixed up?

A: Because Oct 31 = Dec 25.

Given this, what day of December will a mathematician mistake October 25 for?

Hint: Number bases .

This problem is part of the set New Year's Countdown 2017 .


The answer is 21.

Steven Yuan by Steven Yuan , 20, USA

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2 solutions

Steven Yuan
Dec 25, 2017

This joke is about number bases. "Oct 31" refers to the number 31 in base 8, since "oct" is short for "octal," or base 8. "Dec 25" refers to the number 25 in base 10, since "dec" is short for "decimal," or base 10. We can confirm that 3 1 8 = 2 5 10 31_8 = 25_{10} :

3 1 8 = 3 × 8 + 1 = 2 5 10 . 31_8 = 3 \times 8 + 1 = 25_{10}.

Thus, "Oct 25," or 2 5 8 , 25_8, in decimal notation is

2 5 8 = 2 × 8 + 5 = 2 1 10 , 25_8 = 2 \times 8 + 5 = \boxed{21_{10}},

which is our final answer.

Merry Christmas to all at Brilliant! I hope you all have a great holiday and wonderful rest of the year! ⌣ ¨ \ddot \smile

1 Helpful 0 Interesting 0 Brilliant 0 Confused
Tommy Li
Dec 25, 2017

2 5 8 = 2 1 10 25_{8} =21_{10}

0 Helpful 0 Interesting 0 Brilliant 0 Confused

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