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    <pre><code>max_x=5 len=max_x^2 middle=ceiling(max_x/2) A=matrix(NA,max_x,max_x) increments=Reduce( f=function(lhs,rhs) c(lhs,(-1)^(rhs/2+1)*rep(1,rhs)), x=2*(1:(max_x)), init=0 )[1:len] idx_x=Reduce( f=function(lhs,rhs) c(lhs,rep(c(TRUE,FALSE),each=rhs)), 1:max_x, init=FALSE )[1:len] increments_x=increments increments_y=increments increments_x[!idx_x]=0 increments_y[idx_x]=0 A[(middle+cumsum(increments_x)-1)*(max_x)+middle+cumsum(increments_y)]=1:(max_x^2) </code></pre> <p>Gives</p> <pre><code>#&gt; A # [,1] [,2] [,3] [,4] [,5] #[1,] 21 22 23 24 25 #[2,] 20 7 8 9 10 #[3,] 19 6 1 2 11 #[4,] 18 5 4 3 12 #[5,] 17 16 15 14 13 </code></pre> <p>Explanation: The vector <code>increments</code> denotes the steps along the path of the increasing numbers. It's either <code>0/+1/-1</code> for unchanged/increasing/decreasing row and column indices. Important here is that these numbers do not differentiate between steps along columns and rows. This is managed by the vector <code>idx_x</code> - it masks out increments that are either along a row (<code>TRUE</code>) or a column (<code>FALSE</code>). The last line takes into account R's indexing logic (matrix index increases along columns).</p> <hr> <p>Edit: As per request of the OP, here some more information about how the <code>increments</code> vector is calculated.</p> <p>You always go two consecutive straight lines of equal length (row-wise or column-wise). The length, however, increases by 1 after you have walked twice. This corresponds to the <code>x=2*(1:(max_x))</code> argument together with <code>rep(1,rhs)</code>. The first two consecutive walks are in increasing column/row direction. Then follow two in negative direction and so on (alternating). This is accounted for by <code>(-1)^(rhs/2+1)</code>.</p>
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    1. COBTW: I have just been googling "spiral on a grid": [Ulam Spiral](http://en.wikipedia.org/wiki/Ulam_spiral) ... interesting
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    2. COI don't understand this part of your code, could you please explain bit more ? 'f=function(lhs,rhs) c(lhs,(-1)^(rhs/2+1)*rep(1,rhs)), x=2*(1:(max_x)), init=0 )[1:len]'
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    3. COBTW: Why the downvote?
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