Henry Ernest Dudeney/Puzzles and Curious Problems/103 - Expressing Twenty-Four/Solution

From ProofWiki
Jump to navigation Jump to search

Puzzles and Curious Problems by Henry Ernest Dudeney: $103$

Expressing Twenty-Four
In a book published in America was the following:
"Write $24$ with three equal digits, none of which is $8$.
(There are two solutions to this problem.)"
Of course, the answers given are $22 + 2 = 24$, and $3 ^3 - 3 = 24$.
Readers who are familiar with the old "Four Fours" puzzle, and others of the same class,
will ask why there are supposed to be only these solutions.
With which of the remaining digits is a solution equally possible?


Solution

For any digit $n$, we have:

$\dfrac n {\cdotp \dot n} = \dfrac n {n / 9} = 9$

where the notation $\cdotp \dot n$ is used to denote $\cdotp nnnnnn \ldots$

Hence:

$\sqrt {\dfrac n {\cdotp \dot n} } = 3$


This gives us:

$\paren {1 + \sqrt {\dfrac 1 {\cdotp \dot 1} } }! = \paren {1 + 3}! = 4! = 24$

where $!$ denotes the factorial sign.


We have an expression for each of $2$ and $3$.


Then:

$\paren {4 + 4 - 4}! = 4! = 24$

as one of many such similar and straightforward expressions.


We continue:

$\paren {5 - \dfrac 5 5}! = 4! = 24$


To accomplish $6$, we use the fact that or any digit $n$, we have:

$\dfrac n {\cdotp n} = \dfrac n {n / 10} = 10$

Hence:

$\paren {\dfrac 6 {\cdotp 6} - 6}! = \paren {10 - 6}! = 24$


Using the above technique, we see:

$\paren {7 - \sqrt {\dfrac 7 {\cdotp \dot 7} } }! = \paren {7 - 3}! = 4! = 24$


$8$ is simple:

$8 + 8 + 8 = 24$


$9$ falls to the factorial technique again:

$\paren {\sqrt 9 + \dfrac 9 9}! = \paren {3 + 1}! = 4! = 24$


Historical Note

Dudeney supplies only the solutions for $1$ and $7$.

He attributes the solution for $7$ to a certain G.P.E. but the identity of this person has not been established.

According to Martin Gardner, the remaining solutions were determined by Victor Meally, but given the technique for the above, they are easily discovered.


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Henry_Ernest_Dudeney/Puzzles_and_Curious_Problems/103_-_Expressing_Twenty-Four/Solution&oldid=561611"