Mark:
TWO THREE FOUR
THREE = 15400
FOUR = 6724
TWO = 137 or 197
I never even considered 197!
TWO is a 3 digit prime number. The possible primes are shown below. We can eliminate any numbers not consisting of unique digits. And further down we find out that T can only be 1, 2 or 7, so we eliminate any numbers not beginning with those numbers. And also further on we find out that O can only be 1, 3 or 7, so we can eliminate any numbers not ending in those.
3 digit primes
103, 107, 127, 137, 157, 163, 167, 173, 193, 197, 241, 251, 257, 263, 271, 281, 283, 293, 701, 743, 751, 761
From this we know:
T = 1, 2, 7,
W = 0, 2, 3, 4, 5, 6, 7, 8, 9
O = 1, 3, 7
FOUR is a perfect square, so below are all the 4 digit perfect squares possible. There are of course quite a lot more 4 digit squares; however I have eliminated a lot of them because we know the second digit can only be a 1, 3, 7 or 9 and all 4 digits of the number must be unique. And further down we find out that R can only be 4 or 9, so we also can eliminate any numbers not ending in those. So this leaves us with these:
4 digit squares
1369, 1764, 2304, 2704, 5184, 5329, 6724
So we know that:
F = 1, 2, 5, 6
O = 1, 3, 7
U = 0, 2, 6, 8
R = 4, 9
THREE is a triangular number, so we are looking for all 5 digit triangular numbers. They are shown below, and again, a lot of numbers have been eliminated if they don't end in the same last 2 digits, e.g. 22 or 55 and/or if the first 3 digits are not all unique. Since the first 5 digit number is when n=141 which gives 10011 and the last 5 digit number is when n=446 which gives 99681, for this I will use the power of the processor So here's a little php script to produce all 5 digit triangular numbers from n=141 to n=446. It also will eliminate any numbers it finds where the last 2 digits are not the same. And it also eliminates any numbers it finds where the first 3 digits are not all unique, and where none of them match the 4th digit.
Code:
PHP:
<?php
for ($two = 141; $two <= 446; $two++)
{
$one = 0.5;
$three = $two + 1;
$answer = ($one * $two) * $three;
settype($answer,string);
if ($answer{3}==$answer{4})
{
$digits=array();
for ($i=0;$i<strlen>
And finally we can also eliminate any answers where the third number is not equal to 4 or 9 which we know are the only values R can be.
5 digit triangular numbers
15400
17955
27966
75466
So this leaves us with only 4 possible numbers.
We know that:
T = 1, 2, 7
H = 5, 7
R = 4, 9
E = 0, 5, 6
Let's now take a look at the triangular numbers:
We try 17955. This won't work because there is not a 4 digit square possible that ends in a 9 and does not contain the digits 1, 7 or 5.
We try 27966. This also won't work because there is not a 4 digit square possible that ends in a 9 and does not contain the digits 2, 7 or 6.
We try 75466. The only possible 4 digit square is 2304. Which means the 3 digit prime must be 7 ? 3. The only value possible for the question mark is a 4, which has already been used, so this won't work.
So lastly, we try THREE as 15400. The only possible 4 digit square is 6724, which means that TW0 must be 1? 7. The only possible value here is 3 for question mark. There are other possibilities; however they won't work because the numbers would have already been used. So there we have it, our final answers:
The number represented by THREE is 15400
The number represented by FOUR is 6724
The number represented by TWO is 137 (or 197)
Congrats on your conundrums Mis, you have kicked it up a notch.
abpephraovpriiaotre
pbrarigcae
