A little (unsolved) Riddle

This week in math and computer science class we were talking about number riddles, where you can find an explicit solution, which works for every row. For example: \[1+4\Rightarrow 5\\2+5\Rightarrow 12\\3+6\Rightarrow 21\\4+7\Rightarrow \boxed { ? } \] Most of you will get \(32\). Because \(1 \cdot 4 +1 = 5\); \(2 \cdot 5 + 2 = 12\); \(3 \cdot 6 +3 = 21\) and finally \(4 \cdot 7 + 4 = 32\). Hence, for every interger \(a\) and every integer \(b\) we can say: \[a+b\Rightarrow a \cdot b + a = a \cdot \left( b+1 \right)\] But an important information is, that such a riddle doesn't have a unique solution!

But after a while, we were talking about the following riddle and nobody of us found a solution. Even after searching in the www nobody found any formula, which works row by row. 1+2⇒212+3⇒363+4⇒434+5⇒?1+2\Rightarrow 21\\ 2+3\Rightarrow 36\\ 3+4\Rightarrow 43\\ 4+5\Rightarrow \boxed { ? } As you can easily see, in the first and third row, they only flipped the digits. Anyway, we found a few solutions, but these solutions said, that every second row is an exception. But this is neither a formula, which will work row by row nor a general formula, which will work for any integer aa and for any integer bb.

Do you have any ideas, which will work?

Until now, my best try was: a+b⇒b⋅10+(a⋅b)mod10a+b \Rightarrow b \cdot 10 + (a\cdot b) mod 10 . But my best try will fail at the third row by one.

#Algebra

Note by CodeCrafter 1
1 year, 7 months ago

No vote yet
1 vote

  Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

  • Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
  • Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
  • Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
  • Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list
  • bulleted
  • list

1. numbered
2. list
  1. numbered
  2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
2 \times 3 2×3 2 \times 3
2^{34} 234 2^{34}
a_{i-1} ai−1 a_{i-1}
\frac{2}{3} 23 \frac{2}{3}
\sqrt{2} 2 \sqrt{2}
\sum_{i=1}^3 ∑i=13 \sum_{i=1}^3
\sin \theta sin⁡θ \sin \theta
\boxed{123} 123 \boxed{123}

Comments

Weil, you could suppose the equation is in the form a+b→xa+yb+z a + b \rightarrow xa+yb+z . However, this has no solutions for (x,y,z) (x,y,z) .

Then, I tried a+b→xab+ya+zb a+b \rightarrow xab+ya+zb and this actually works. We get the solution (x,y,z)=(−4,33,−2) (x,y,z)=(-4,33,-2) . For 4+5 4+5 this actually results in 42 42 and then decreases even further.

Henry U - 1 year, 7 months ago

Log in to reply

Wow, that's amazing. Thank you!

Funfact: I think it's a divine sign that the answer is 42 ;)

CodeCrafter 1 - 1 year, 7 months ago
×

Problem Loading...

Note Loading...

Set Loading...