Don't miss the beautiful colours of the rainbow while you're looking for the pot of gold at the end of it. I only mention it because; ideas are like children-no matter how much you admire someone else's, you can't help liking your own best.
Mark:
GRID
125
289
127
This is definitely a case of the explanation being more complicated than the result, but it gives a small insight into the reasoning to produce a unique result, and speaks volumes about those capable of such feats.
We have 2 scenarios.
Scenario 1 - Using parts A, B, and C
From that we can see that we have 9 grids to fill, with 3 answers. Since the maximum answer can be 3 digits long, we know that the answer to all three parts will be 3 digits, which also tells us that the only possible starting position combinations are 1, 4, 7 or 1, 2, 3.
We know that A can start from 1, 4
We know that B can start from 1, 8
We know that C can start from 2, 3, 5, 7
Now to work out the correct grid position to use. So we try with A starting from 1. If A goes downwards, then we need to also start from 2 and 3. This won't be possible because B cannot start form either of those. If A goes across, then we have to also start from 4 and 7. Again this won't work because B cannot start from either of those. So to conclude from this, A must start from grid 4, and also must go across the grid. We are left with 1 and 7 to start from. And it can easily be seen that B will have to start from grid position 1 and C will have to start from grid position 7.
BBB
AAA
CCC
Scenario 2 - Using parts A, B, C and D
We have 4 answers to fill 9 grid positions. So 3 of the answers will only be 2 digits long, and 1 answer will be 3 digits long. Let's have a look at possible grid layouts.
We know that A can start from 1, 4
We know that B can start from 1, 8
We know that C can start from 2, 3, 5, 7
We know that D can start from 2, 4, 6, 8
I found 6 possible grid positions for scenario 2, shown below:
--1--------2---------3-------4-------5--------6--
AAA --- BBB --- BBB --- BBB --- ADD --- ADC
DCC--- ACC --- ACD ---AAD --- ACC --- ADC
DBB --- ADD --- ACD ---CCD ---ABB --- ABB
Lets have a look at each of these, comparing each grid with that of scenario 1.
Number 1: We have AAA equaling BBB from the scenario 1 grid. This is possible, and can only equal 729, since that is the only answer that is in both A and B. We can also see that DCC equals AAA from the scenario 1 grid. So the last two digits of AAA have to equal a two digit answer to C. Well there aren't any, so this rules out grid 1 as the final grid to use.
Number 2: We have BBB equaling BBB. This is possible, and so BBB could actually be any 3 digit answer to B it wanted. However, looking at the next line, we have ACC equalling AAA. From grid 1 we already know this won't work since there are no two digit answers to C that end a 3 digit answer to A. So this is also not the correct grid to use.
Number 3: Again we have BBB equalling BBB so that is possible. Next I'm going to look at DD which I know must be an even number, and hence its last digit must be an even number. And comparing that to scenario 1 grid, we have CCC going across and its last digit is the last digit of DD. Now CCC must be a prime number, so it will never end in an even number. So this means this grid is not the correct one. Are you still reading? Well done, only 3 more to look at!
Number 4: Again, we have BBB equalling BBB. And again though, we have DD in the same place as number 3, and so we already know that this grid is not correct.
Number 5: Straight away we should be able to see this won't work. Like number 2, we have ACC equalling AAA which we know isn't possible.
Number 6: The last grid, and so by the process of elimination, unless there are more grids, this one should work. Guess what, it will. Let's take a look.
Scenario 1 Grid ------- Scenario 2 Grid
------ BBB -------------------- ADC -----
------ AAA -------------------- ADC -----
------ CCC -------------------- ABB -----
Difficult one on where to start for this, but ABB equals CCC looks like a good one to start with. So we have the last two digits of CCC equalling BB. BB can only be either 27 or 64. Let's try 64. Straight away we know there are no CCC answers ending in a 4 because it is even, and CCC must be a prime number. So here's a starting point. BB equals 27.
Now we look at any CCC answers ending in 27. There are two possibilities. CCC = 127 or 727. So we try with 727. This makes our AAA answer in scenario 2 end with a 7. Well this won't work because there are no AAA answers ending in a 7. Therefore, we now know that CCC must equal 127.
Lets have a look at our grids again with the numbers we know filled in.
Scenario 1 Grid ------- Scenario 2 Grid
------ BBB -------------------- ADC -----
------ AAA -------------------- ADC -----
------ 127 --------------------- 127 -----
Now we know that AAA in scenario 2 must end in a 1. So AAA = 121, 361, 441, 841, or 961. We also know that whatever AAA begins with, BBB in scenario 1 will also have to begin with it. So we can actually rule out a few numbers in this. BBB can only begin with a 1, 2, 3, 5 or 7. Therefore AAA in scenario 2 can only equal 121 or 361. And from that BBB in scenario 1 can only equal 125 or 343. Now if BBB is 125, then CC in scenario 2 must begin with a 5, so it could be 53 or 59. Now AAA in scenario 1 must end with a 3 or 9. It will never end in a 3, therefore CC in this case would have to be 59. But if BBB is 343, then CC in scenario 2 must begin with a 3, so it could be 31 or 37. Now AAA in scenario 1 must end with a 1 or 7. It will never end in a 7, therefore CC in this case would be 31. Therefore, we now know that CC can only be 59 or 31.
So now AAA in scenario 1 must end with either a 9 or 1. Let's have a look at possible answers to AAA. We have 121, 169, 289, 361, 441, 529, 729, 841, 961.
Now going back to what we worked out earlier. AAA in scenario 2 can equal 121 or 361. Now AAA in scenario 1 will have to begin with the second digit of AAA from scenario 2. So either a 2 or 6. Looking at AAA's possibilities for scenario 1, there are none beginning with a 6. Therefore, AAA in scenario 2 must be 121 and thus AAA in scenario 1 must be 289. LFirst things first, have a look at what we know for fact. The post above shows all possible starting points for each part of the question which there are 4 of: A, B, C, and D. It also shows all the possible answers. Other things we know for fact are:
We have 2 scenarios.
Scenario 1 - Using parts A, B, and C
From that we can see that we have 9 grids to fill, with 3 answers. Since the maximum answer can be 3 digits long, we know that the answer to all three parts will be 3 digits, which also tells us that the only possible starting position combinations are 1, 4, 7 or 1, 2, 3.
We know that A can start from 1, 4
We know that B can start from 1, 8
We know that C can start from 2, 3, 5, 7
Now to work out the correct grid position to use. So we try with A starting from 1. If A goes downwards, then we need to also start from 2 and 3. This won't be possible because B cannot start form either of those. If A goes across, then we have to also start from 4 and 7. Again this won't work because B cannot start from either of those. So to conclude from this, A must start from grid 4, and also must go across the grid. We are left with 1 and 7 to start from. And it can easily be seen that B will have to start from grid position 1 and C will have to start from grid position 7.
BBB
AAA
CCC
Scenario 2 - Using parts A, B, C and D
We have 4 answers to fill 9 grid positions. So 3 of the answers will only be 2 digits long, and 1 answer will be 3 digits long. Let's have a look at possible grid layouts.
We know that A can start from 1, 4
We know that B can start from 1, 8
We know that C can start from 2, 3, 5, 7
We know that D can start from 2, 4, 6, 8
I found 6 possible grid positions for scenario 2, shown below:
--1--------2---------3-------4-------5--------6--
AAA --- BBB --- BBB --- BBB --- ADD --- ADC
DCC--- ACC --- ACD ---AAD --- ACC --- ADC
DBB --- ADD --- ACD ---CCD ---ABB --- ABB
Lets have a look at each of these, comparing each grid with that of scenario 1.
Number 1: We have AAA equaling BBB from the scenario 1 grid. This is possible, and can only equal 729, since that is the only answer that is in both A and B. We can also see that DCC equals AAA from the scenario 1 grid. So the last two digits of AAA have to equal a two digit answer to C. Well there aren't any, so this rules out grid 1 as the final grid to use.
Number 2: We have BBB equaling BBB. This is possible, and so BBB could actually be any 3 digit answer to B it wanted. However, looking at the next line, we have ACC equalling AAA. From grid 1 we already know this won't work since there are no two digit answers to C that end a 3 digit answer to A. So this is also not the correct grid to use.
Number 3: Again we have BBB equalling BBB so that is possible. Next I'm going to look at DD which I know must be an even number, and hence its last digit must be an even number. And comparing that to scenario 1 grid, we have CCC going across and its last digit is the last digit of DD. Now CCC must be a prime number, so it will never end in an even number. So this means this grid is not the correct one. Are you still reading? Well done, only 3 more to look at!
Number 4: Again, we have BBB equalling BBB. And again though, we have DD in the same place as number 3, and so we already know that this grid is not correct.
Number 5: Straight away we should be able to see this won't work. Like number 2, we have ACC equalling AAA which we know isn't possible.
Number 6: The last grid, and so by the process of elimination, unless there are more grids, this one should work. Guess what, it will. Let's take a look.
Scenario 1 Grid ------- Scenario 2 Grid
------ BBB -------------------- ADC -----
------ AAA -------------------- ADC -----
------ CCC -------------------- ABB -----
Difficult one on where to start for this, but ABB equals CCC looks like a good one to start with. So we have the last two digits of CCC equalling BB. BB can only be either 27 or 64. Let's try 64. Straight away we know there are no CCC answers ending in a 4 because it is even, and CCC must be a prime number. So here's a starting point. BB equals 27.
Now we look at any CCC answers ending in 27. There are two possibilities. CCC = 127 or 727. So we try with 727. This makes our AAA answer in scenario 2 end with a 7. Well this won't work because there are no AAA answers ending in a 7. Therefore, we now know that CCC must equal 127.
Lets have a look at our grids again with the numbers we know filled in.
Scenario 1 Grid ------- Scenario 2 Grid
------ BBB -------------------- ADC -----
------ AAA -------------------- ADC -----
------ 127 --------------------- 127 -----
Now we know that AAA in scenario 2 must end in a 1. So AAA = 121, 361, 441, 841, or 961. We also know that whatever AAA begins with, BBB in scenario 1 will also have to begin with it. So we can actually rule out a few numbers in this. BBB can only begin with a 1, 2, 3, 5 or 7. Therefore AAA in scenario 2 can only equal 121 or 361. And from that BBB in scenario 1 can only equal 125 or 343. Now if BBB is 125, then CC in scenario 2 must begin with a 5, so it could be 53 or 59. Now AAA in scenario 1 must end with a 3 or 9. It will never end in a 3, therefore CC in this case would have to be 59. But if BBB is 343, then CC in scenario 2 must begin with a 3, so it could be 31 or 37. Now AAA in scenario 1 must end with a 1 or 7. It will never end in a 7, therefore CC in this case would be 31. Therefore, we now know that CC can only be 59 or 31.
So now AAA in scenario 1 must end with either a 9 or 1. Let's have a look at possible answers to AAA. We have 121, 169, 289, 361, 441, 529, 729, 841, 961.
Now going back to what we worked out earlier. AAA in scenario 2 can equal 121 or 361. Now AAA in scenario 1 will have to begin with the second digit of AAA from scenario 2. So either a 2 or 6. Looking at AAA's possibilities for scenario 1, there are none beginning with a 6. Therefore, AAA in scenario 2 must be 121 and thus AAA in scenario 1 must be 289. Let's have a look at this on the grids:
Scenario 1 Grid ------- Scenario 2 Grid
------ 1BB -------------------- 1DC -----
------ 289 --------------------- 289 -----
------ 127 --------------------- 127 -----
Now straight away we see that BBB in scenario 1 begins with a 1, and so it can only equal 125. And that actually completes the grid!
Scenario 1 Grid ------- Scenario 2 Grid
------ 125 --------------------- 125 -----
------ 289 --------------------- 289 -----
------ 127 --------------------- 127 -----
To double check this is all correct:
Scenario 1
BBB = 125
AAA = 289
CCC = 127
Scenario 2
AAA = 121
DD = 28
CC = 59
BB = 27
All of these answers work out correct. And so the final grid looks like this: Let's have a look at this on the grids:
Scenario 1 Grid ------- Scenario 2 Grid
------ 1BB -------------------- 1DC -----
------ 289 --------------------- 289 -----
------ 127 --------------------- 127 -----
Now straight away we see that BBB in scenario 1 begins with a 1, and so it can only equal 125. And that actually completes the grid!
Scenario 1 Grid ------- Scenario 2 Grid
------ 125 --------------------- 125 -----
------ 289 --------------------- 289 -----
------ 127 --------------------- 127 -----
To double check this is all correct:
Scenario 1
BBB = 125
AAA = 289
CCC = 127
Scenario 2
AAA = 121
DD = 28
CC = 59
BB = 27
All of these answers work out correct. And so the final grid looks just like Mark's:
1 2 5
2 8 9
1 2 7
ccnnneehpsepiasadst?
(It's 2 words from a famous book!)
Ok, it's my bad!
I read it as; the Title of a famous book, not two words taken from its pages. However there are few questions here that go unanswered, so I echo TTH comments; but to make it into an anagram was sneaky.
TTH writes,
"Hey try..........do you know these guys?"
I don't wanna bang my own drumÂ…..but they look like distant relatives.
APREPSROPPORINATSEE
EIPNT
