@OP so the constitution guarantees the right of gun ownership for protection or defence. How then does it protect the citizen from gun ownership for murder, where there is no thought of defence, just of murder? Constitutional right to buy a murder weapon having murderous intent? Oh, but then that would be excusable madness, wouldn't it?
I was thinking about the diagonal abcd with a+c=b+d=17 as ted mentioned, and it wasn't immediately obvious to me but it is simple algebra. Each solution has variants based on rotation and reflection or the symmetry group of a square The image is of a spreadsheet in which ordering the bcde or 2481 can generate all variants which have R1C1 = a =1 (or R1C4=a=1) - there should be 16.
Personally I believe in nothing and need nothing. A belief which will never change. I would simply paraphrase Descartes as 'sum, ergo sum' being unsure about the 'thinking/cogito' aspect. Allow me to go back to my current Kenken problem and 'think' about it!
@merlot @ted used matrix algebra, and that allows programming - but there are 20 equations and 18 variables, 2 equations need elimination. I haven't solved it - very time consuming by logic.
The four corners and the central quadrant are derivable, so that reduces the equations to 4+4+8+2=18 Still 2 too many, and the diagonal constraints are used in the proof, yet they are the only candidates. So perhaps we need to show the other constraints imply the diagonals.
@OP I get more than enough of the 'hi!' mail, but the number who exceed half my age, are not strikingly good looking, and do not live in America is in single digit territory. Usually I am about grandpa age. The number who respond with appropriate interests and age I can count on my thumbs.
I am not teaching him matrix inversion - you'd accuse me of being Charles Dodgson! I'm doing it by pure logic and elimination. Notice how the 23-11 diagonal pattern propagates as a result of the initial clues. The 2x2 square constraint in fact makes it considerably easier, not harder.
@merlot think of the 2x2 square as defined by its top left corner. Then every square except column 4 and row 4 has a corresponding 2x2. This is the 3x3 square R1C1 - R3C3 - 9 squares.
Haha I see 7 is too easy. Here's a level 9, which is getting pretty fiendish! I left a shot in of the browser tab so you can see the url if you care to visit the website.
@ted for the 4x4 there are 4+4+2+9 = 19 equations for 16 variables so there is clear redundancy, non-independency. But yes I missed the requirement for all 9 2x2 squares! Back to the drawingboard, but it still makes bot the boy and myself think intensely.
There is no requirement for the R1C2xR2C3 square to add to 34, nor for R2C1xR3C3; the Four corners and the central square yes. The requirement is clearly stated in the image.
When n is a multiple of 4 there are additional properties. Not sure how Ramanujan comes into this however, but it is amusing.
Merlot the quadrants in 4x4 and 8x8 magic squares are also equal to the sum - the corner 2x2 squares also sum to 34 in the 4x4 case. Hence you notice the 17 but it is in two rows. It is not a requirement of the puzzle.
no there are frequently two solutions where you may have a rectangle a or b | b or a --------|--------- c or d | d or c Where you can swap a/c b/d if a+c = b+d then row and column sums are unaffected if located appropriately The second solution I provided is a valid second solution. Yours was valid too. The linear equations do not express the full problem. You have n^2 variables with 2n+4+c equations. Morever they are not linearly independent. I've forgotten the terminology for problems involving one-one pigeon-holing.
yes - look at 1 and 3 - they cannot go in rows 2 or 4, and they cannot go in the same column. Since column givens add to 18 and we need 16, so (1,15), (3,13) occupy 4 squares in two columns. The first trial of 1 at R1C1 turns up trumps. Sometimes there are dual solutions involving exchanging ab/ba pairs and symmetry. Using linear algebra doesn't seem to work since the equations are not independent. 9,8,3,14 12,5,2,15 6,11,16,1 7,10,13,4 seems to be such a dual
Since the four corners add to 34, that tells us that R1C1+R4C1 = 16, so R2C1 = 34 - 16 - 6 and that is the last even number. All the others remaining are odd, two each per column and each column pair adds to 16...
Someone here says you grow out of friendships? I have very few - one hand is enough. But in fact I find you grow into them and they are life-long, they are truly cherished.
Watching people is not so strange: On Sunday nights my parents used to take (as a kind of relaxation) a drive to city on Sunday night, midnight more or less, just to get the Sunday-night paper, and watch the weird locals in Kings-Cross
A dictionary the doesn't have derision? Wow! As an aside my C-E dictionary PLECO doesn't have kookaburra. Whoops it's been updated and now it does! It certainly has derision.
online today!FargoFan Blog Comments (1,859)
RE: The right to bear arms
@OP so the constitution guarantees the right of gun ownership for protection or defence. How then does it protect the citizen from gun ownership for murder, where there is no thought of defence, just of murder? Constitutional right to buy a murder weapon having murderous intent? Oh, but then that would be excusable madness, wouldn't it?