# Number Spiral Problem

A number spiral is an infinite grid whose upper-left square has number 1. Here are the first five layers of the spiral:

Your task is to find out the number in row y and column x.

Example:

Input: 2 3
Out  : 8
Input: 1 1
Out  : 1
Input: 4 2
Out  : 15


Note:

1. Any programming language is allowed.
2. This is a challenge so shortest code wins.
3. Best of Luck!
• @WW What does that mean ? Aug 17, 2018 at 16:15
• It looks like your inputs are 1 indexed (coordinates start at 1,1) (although this has to be intuited from the test cases) can we use 0 indexing (coordinates start at 0,0)? Aug 17, 2018 at 16:26
• What is the reasoning for this? Aug 17, 2018 at 16:30
• I think it's absolutely fine for the coordinates to start at (1, 1), especially if the program is posted that way on CSES, and the OP doesn't need to justify this. I think golfers here are getting a little too used to somewhat arbitrary freedoms.
– Lynn
Aug 18, 2018 at 12:57
• @Lynn I second that Aug 18, 2018 at 12:58

## C (gcc),  44  43 bytes

f(x,y,z){z=x>y?x:y;z=z*z-~(z%2?y-x:x-y)-z;}


Try it online!

The spiral has several "arms":

12345
22345
33345
44445
55555


The position $(x, y)$ is located on arm $\max(x, y)$ (assigned to variable z). Then, the largest number on arm $n$ is $n^2$, which alternates between being in the bottom left and top right position on the arm. Subtracting $x$ from $y$ gives the sequence $-n+1, -n+2, \ldots, -1, 0, 1, \ldots, n-1, n-2$ moving along arm $n$, so we choose the appropriate sign based on the parity of $n$, adjust by $n-1$ to get a sequence starting at 0, and subtract this value from $n^2$.

Thanks to Mr. Xcoder for saving a byte.

• f(x,y,z){z=x>y?x:y;z=z*z-~(z%2?x-y:y-x)-z;} saves 1 byte. Aug 17, 2018 at 15:52
• @Mr.Xcoder Neat trick, thanks! Aug 17, 2018 at 15:55
• Crossed out 44 is still regular 44 ;( Aug 17, 2018 at 16:11
• @RobertS. Yes, that is what the function I defined does (in the Code section on TIO). For instance, f(1, 1) returns the value 1. The Footer section loops through x=1 through 5 and y=1 through 5, calls the function for all such values, and prints its output in a grid, to demonstrate that the function is correct for all inputs shown in the question. Aug 17, 2018 at 19:38
• @Agile_Eagle The function does return the number (it couldn't output the spiral - it doesn't even have any loops!). Aug 17, 2018 at 19:40

## Python,  54  50  49 bytes

def f(a,b):M=max(a,b);return(a-b)*(-1)**M+M*M-M+1


-4 bytes thanks to @ChasBrown

-1 bytes thanks to @Shaggy

Try it Online!

First time golfing! I'm more than aware this is not optimal, but whatever.

Essentially runs on the same principle as @Doorknob C code.

• Welcome to PPCG! In this case you can save 4 bytes using the def f(a,b): approach, see here. Aug 17, 2018 at 21:21
• @ChasBrown Very interesting, thank you! Aug 17, 2018 at 21:21
• @Shaggy Thank you! I've posted a few challenges, but never been good enough to golf Aug 17, 2018 at 22:32
• In that case, then, welcome to Golf! :) I'm not a Python guy but I'm pretty sure M**2 can be replaced with M*M. Aug 17, 2018 at 22:36
• @Shaggy Thank you! Will fix right now Aug 17, 2018 at 22:37

# MATL, 15 bytes

X>ttq*QwoEqGd*+


### How?

Edit: Same technique as @Doorknob's answer, just arrived at differently.

The difference between the diagonal elements of the spiral is the arithmetic sequence $0, 2, 4, 6, 8, \ldots$. Sum of $n$ terms of this is $n(n - 1)$ (by the usual AP formula). This sum, incremented by 1, gives the diagonal element at position $(n, n)$.

Given $(x, y)$, we find the maximum of these two, which is the "layer" of the spiral that this point belongs to. Then, we find the diagonal value of that layer as $v = n(n-1) + 1$. For even layers, the value at $(x, y)$ is then $v + x - y$, for odd layers $v - x + y$.

X>        % Get the maximum of the input coordinates, say n
ttq*      % Duplicate that and multiply by n-1
Q         % Add 1 to that. This is the diagonal value v at layer n
wo        % Bring the original n on top and check if it's odd (1 or 0)
Eq        % Change 1 or 0 to 1 or -1
Gd        % Push input (x, y) again, get y - x
*         % Multiply by 1 or -1
% For odd layers, no change. For even layers, y-x becomes x-y
+         % Add that to the diagonal value v
% Implicit output


Alternate 21 byte solution:

Pdt|Gs+ttqq*4/QJb^b*+


Try it online!
Collect and print as a matrix
From the above, we know that the function we want is

$$f = m * (m - 1) + 1 + (-1)^m * (x - y)$$

where $m = max(x, y)$.

Some basic calculation will show that one expression for max of two numbers is

$$m = max(x, y) = \frac{x + y + abs(x - y)}{2}$$

Plugging one into another, we find that one alternate form for $f$ is:

$$f = (x-y)\cdot i^{k} + \frac{1}{4}((k-2)\cdot k) + 1$$

where $k = abs(x-y) + x + y$.

This is the function the solution implements.

# Japt, 1614 10 bytes

(Originally) Adapted from Doorknob's solution over a few beers.

wV
nU²ÒNra


Try it

wV\nnU²ÒNra     :Implicit input of integers N=[U=y, V=X]
wV              :Maximum of U & V
\n            :Reassign to U, leaving the value in N unchanged
n           :Subtract U from
U²         :  U squared
Ò        :  Subtract the bitwise NOT of
Nr      :  N reduced by
a     :    Absolute difference


# Pyth, 20 bytes

A~Qh.MZQh-+*-GH^_1Q*


Test suite

An almost literal translation of Rushabh Mehta's answer.

Explanation:
A~Qh.MZQh-+*-GH^_1Q*    | Full code
A~Qh.MZQh-+*-GH^_1Q*QQQ | Code with implicit variables filled
| Assign Q as the evaluated input (implicit)
A                       | Assign [G,H] as
~Q                     |  Q, then assign Q as
h.MZQ                |   Q's maximal value.
| Print (implicit)
h-+*-GH^_1Q*QQQ |  (G-H)*(-1)^Q+Q*Q-Q+1


# Jelly, 13 bytes

»Ḃ-*×_‘+»×’\$¥


Try it online!

Uses Doorknob's method. Way too long.

• Alternative: »Ḃ-*×_‘+»²_»ʋ Aug 17, 2018 at 16:41

# Jelly, 13 12 bytes

ṀḂḤ’×I+²_’ṀƲ


Try it online!

Computes the diagonal term with ²_’Ṁ and adds/subtracts to the correct index value with ṀḂḤ’×I.

# Brain-Flak, 76 bytes

((({}<>))<>[(({}))]<{({}[()])<>}>)<>{}((){({}[()])({})<><([{}])><>}{}<>{}<>)


Try it online!

# 05AB1E, 12 11 bytes

ZÐ<*>ŠGR}¥+


-1 byte thanks to @Emigna changing Èi to G.

Port of @sundar's MATL answer, so make sure to upvote him!

Explanation:

Z              # Get the maximum of the (implicit) input-coordinate
#  i.e. [4,5] → 5
Ð             # Triplicate this maximum
<            # Decrease it by 1
#  i.e. 5 - 1 → 4
*           # Multiply it
#  i.e. 5 * 4 → 20
>          # Increase it by 1
#  i.e. 20 + 1 → 21
Š         # Triple swap the top threes values on the stack (a,b,c to c,a,b)
#  i.e. [4,5], 5, 21 → 21, [4,5], 5
G }      # Loop n amount of times
R       #  Reverse the input-coordinate each iteration
#   i.e. 5 and [4,5] → [5,4]→[4,5]→[5,4]→[4,5] → [5,4]
¥     # Calculate the delta of the coordinate
#  [5,4] → [1]
+    # And add it to the earlier calculate value (output the result implicitly)
#  21 + [1] → [22]

• Èi could be G. Aug 20, 2018 at 10:39
• @Emigna Oh smart, thanks! :D Aug 20, 2018 at 11:01

# Pascal (FPC), 90 bytes

uses math;var x,y,z:word;begin read(x,y);z:=max(x,y);write(z*z-z+1+(1and z*2-1)*(y-x))end.


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Port of Doorknob's answer, but sundar's answer gave me idea for z mod 2*2-1 which I transformed into 1and z*2-1 to remove space.

# Mathematica 34 bytes

x = {5, 8};


so:

m = Max[x];
Subtract @@ x (-1)^m + m^2 - m + 1


(*

54

*)

# Julia 1.0, 35 bytes

x\y=(m=max(x,y))*~-m+1+(-1)^m*(x-y)


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# JavaScript (ES6), 46 bytes

f=(r,c,x)=>r<c?f(c,r,1):r%2-!x?r*r-c+1:--r*r+c


# Java (JDK 10), 39 bytes

x->y->(y-x)*((y=x>y?x:y)%2*2-1)+y*y-y+1


Try it online!

# Rockstar, 127118 115 bytes

listen to Y