For the math class I'm teaching this summer, I've tried to inject a bit of puzzling into the mix to get more excitement out of the students. One type of puzzles I've come across is called algebraic puzzles, where you are asked to find digits to directly substitute for the letters of a phrase or statement, where the mathematical statement is also correct. Here's an example.
Substitute unique digits for the following statement such that it is mathematically correct:
CL = LOGIC
I wrote a little Javacript script to calculate the solution using random digits generated for each of the letters until the mathematical statement was true. The three possibilities are:
\[1^0 = 00001\] \[9^5 = 59049\] \[5^7 = 78125\]Of course, only the last solution satisfies the puzzle statement, but I think that this is a great example of thinking algorithmically. Hopefully, one can see how this sort of script could solve a whole series of these types of puzzles without the necessity of trying endless combinations of digits by hand. The script takes less than a second to run each time, even though it runs through potentially countless random calculations of digits. This shows how computers can greatly enhance our ability to solve problems, if we can think in ways that allow us to program them to generally solve a problem given a specific instance of it.
Below this is the script (without the output line) and the output of this particular run of the script:
do
{
a = Math.floor(Math.random() * 10);
b = Math.floor(Math.random() * 10);
c = Math.floor(Math.random() * 10);
d = Math.floor(Math.random() * 10);
f = Math.floor(Math.random() * 10);
}
while (a ** b != 10000 * b + 1000 * c + 100 * d + 10 * f + a);
Here's a challenge of another of these so-called algebraic puzzles with the same theme.
Solve the following (each letter is a separate digit):
(L+O+G+I+C)3 = LOGIC
Here are the answers I got using a similar Javascript script as above:
\[(1+7+5+7+6)^3 = 17576\] \[(0+4+9+1+3)^3 = 04913\] \[(1+9+6+8+3)^3 = 19683\] \[(0+5+8+3+2)^3 = 05832\] \[(0+0+5+1+2)^3 = 00512\]Only the third answer matches the puzzle criteria exactly. The rest either have non-unique digits or have zeroes in the first position of a number or both. There are more answers that match the criterion I set in the script, but they are trivial, like \(0^3=0\) and \(1^3=1\), so I will not present them above, but are possibilities if you relax the rules of the puzzle.
I tried using the same Javascript approach for this problem:
Substitute unique digits (0-9) for the letters in these two addition problems:
ONE + ONE = TWO and ONE + FOUR = FIVE
But for some reason, the necessity of the two equations being satisfied at the same time didn't allow this method to work. I leave it to the reader to solve this problem, via a script or just trial and error and pencil and paper.