# এই সংখ্যাটি কি ত্রিভুজাকার?

33

## চ্যালেঞ্জ

ধনাত্মক পূর্ণসংখ্যা দেওয়া, এটি ত্রিভুজাকার সংখ্যা কিনা তা নির্ধারণ করুন এবং ততক্ষণে দুটি ধ্রুবক, স্বতন্ত্র মানগুলির মধ্যে একটি আউটপুট।

### সংজ্ঞা

একটি ত্রিকোণ সংখ্যা একটি সংখ্যা যে পরপর ধনাত্মক পূর্ণসংখ্যা এর সমষ্টি হিসাবে প্রকাশ করা যেতে পারে, 1. থেকে শুরু তারা সূত্র সঙ্গে প্রকাশ করা যেতে পারে n(n + 1) / 2, যেখানে nকিছু পূর্ণসংখ্যা ইতিবাচক হয়।

## পরীক্ষার মামলা

Truthy:

1
3
6
10
15
21
55
276
1540
2701
5050
7626
18915
71253
173166
222111
303031
307720
500500
998991


Falsy:

2
4
5
7
8
9
11
16
32
50
290
555
4576
31988
187394
501500
999999


## বিধি

• আপনার এন্ট্রি কোনও ফাংশন বা কোনও প্রোগ্রাম হতে পারে।
• আপনি ধরে নিতে পারেন যে ইনপুটটি 10 6 এর নিচে একটি ধনাত্মক পূর্ণসংখ্যা ।
• দুটি বিভাগকে আলাদা করতে আপনাকে অবশ্যই দুটি ধ্রুবক, স্বতন্ত্র আউটপুট বেছে নিতে হবে।

এটি , তাই প্রতিটি ভাষার বাইটের মধ্যে সংক্ষিপ্ততম কোডটি জয়ী।

আপনি শূন্য অন্তর্ভুক্ত করেননি কেন?
নিল

1
@Neil I wanted to minimize the number of possible edge cases, and handling zero is one of them that I felt wasn't too important. Do you think it would have been better if zero needed to be handled? (The Jelly answer currently fails on zero, for instance)
ETHproductions

উত্তর:

21

EDIT:

• -1 byte: @xnor got rid of parentheses with a $. An anonymous function taking an Int and returning a Char. Output is '1' for triangular numbers and '0' for others. (!!)$show.(10^)=<<[0..]

Try it online!

• Use as ((!!)$show.(10^)=<<[0..]) 998991. • Generates the numbers 1, 10, 100, 1000, ..., converts those to strings, and concatenates them. Then indexes into the resulting infinite string "1101001000100001000001000000... 6 An imaginative method! You can save a byte with (!!)$show.(10^)=<<[0..].
xnor

20

# Python, 24 bytes

lambda n:(8*n+1)**.5%1>0

Try it online!

Outputs False for triangular numbers, True for the rest. Checks if 8*n+1 is a perfect square. Python will take perfect squares to exact integer floats no matter how large, so there are no floating-point issues.

3
(1<<10000)**.5: OverflowError: int too large to convert to float
isaacg

@isaacg The challenge doesn't require inputs that large: "You may assume that the input is a positive integer under 10^6"
trichoplax

1
@trichoplax I think I was disputing xnor's claim in the text. The submission is fine, I agree.
isaacg

13

# Jelly, 4 bytes

R+\ċ


Try it online!

### How?

R+\ċ - Main link: n
R    - range(n)   -> [1,2,3,...,N]
\  - cumulative reduce by:
ċ - count occurrences of right (n) in left -> 1 if triangular, 0 otherwise


I'm surprised cumulative reduce doesn't automatically make a range. Is there a design choice behind this?
ETHproductions

I am not 100% sure, but I think it would (at least currently) need to by the dyadic operation being reduced over that would cause a range to be made.
Jonathan Allan

...actually even that does not seem to apply (e.g. this vs this. It seems the quicklink implementation overrides such that the iterable does not make a range even if the dyadic operation defines it to do so for an argument. Pinged Dennis to field this one :)
Jonathan Allan

@ETHproductions / and \ probably were among the first five quicks to be implemented, predating the idea to cast integer arguments to range.
Dennis

13

## Retina, 10 bytes

(^1|1\1)+$ Input is in unary. Output is 0 or 1. Try it online! (As a test suite that does decimal-to-unary conversion for convenience.) ### Explanation This is the most basic exercise in forward-references. Most people are familiar with backreferences in regex, e.g. (.)\1 to match a repeated character. However, some of the more advanced flavours allow you to use a backreference before or inside the group it's referring to. In that case, it's usually called a forward-reference. This can make sense if the reference is repeated. It might not be well defined on the first iteration, but on subsequent iterations, the later or surrounding group has captured something and can be reused. This is most commonly used to implement recurrent patterns on unary strings. In this case, we try to match the input as the sum of consecutive integers: ( # This is group 1, which we'll repeat 1 or more times. ^1 # Group 1 either matches a single 1 at the beginning of the string. | # or 1\1 # It matches whatever the previous iteration matched, plus another # 1, thereby incrementing our counter. # Note that the first alternative only works on the first iteration # due to the anchor, and the second alternative only works *after* # the first iteration, because only then the reference is valid. )+$        # Finally, we make sure that we can exactly hit the end of the
# string with this process.


1
Why doesn't (^|1\1)+$ work? Leaky Nun 3 @LeakyNun regex engines have an optimisation that they stop repeating a group if it was empty n times where n is the minimum of the quantifier you're using (in your case 1; if the minimum was 0, it would be tried once anyway). If you change the + to {2,}, it should work. This optimisation prevents infinite loops but it's also the only thing that keeps .NET regex from being Turing-complete on its own. Martin Ender This just saved me 70 bytes: codegolf.stackexchange.com/a/118387 Neil Make that 74 bytes, thanks to \G! Neil 8 ## Mathematica, 16 bytes OddQ@Sqrt[1+8#]&  Essentially a port of xnor's Python solution. Outputs True for triangular numbers, False otherwise. 7 ## JavaScript (ES6), 30 27 bytes Saved 2 bytes thanks to kamoroso94 f=(n,k)=>n>0?f(n+~k,-~k):!n ### Test cases ### Non-recursive version (ES7), 19 bytes Port of Adnan's answer. x=>(8*x+1)**.5%1==0 Only seeing now that you edited the 19 byte solution into your answer a few minutes before I posted mine. Should I delete mine? What's the generally accepted etiquette on that? Shaggy 1 @Shaggy I don't think it's a real problem here. My 'main' answer really is the recursive one. Arnauld Reduce to 28 bytes with f=(n,k=1)=>n>0?f(n-k,k+1):!n? kamoroso94 1 @kamoroso94 Thanks! Updated. And a third byte was saved by omitting the initialization of k. Arnauld Elegant use of bitwise NOT as an incrementor for an initially-undefined value; your edit was a pleasure to read after I independently arrived at your prior solution. apsillers 6 # CJam, 11 bytes ri2*_mQ_)*=  Outputs 1 for triangular, 0 otherwise. Try it online! ### Explanation Consider input 21. ri e# Input integer. STACK: 21 2* e# Multiply by 2. STACK: 42 _ e# Duplicate. STACK: 42, 42 mQ e# Integer square root. STACK: 42, 6 _) e# Duplicate, increment. STACK: 42, 6, 7 * e# Multiply. STACK: 42, 42 = e# Equal? STACK: 1  6 # Brain-Flak, 40 bytes (([{}](((()))<>))<>){<>({}({}({})))}{}{}  Wheat Wizard and I had a duel over this question. When we decided to post our solutions we were tied at 42 bytes, but I found a 2 byte golf of his solution. We decided that would count as the tie breaker (my solution is below). Try it online! ### Explanation: # Set up the stacks like this: -input 1 -input 1 1 (([{}](((()))<>))<>) ^ # Output 1 for triangular and 0 for non-triangular {<>({}({}({})))}{}{}  For a full explanation please see Wheat Wizard's answer. # Brain-Flak, 42 bytes (([({})])<>){(({}())<>{}({})){((<>))}{}{}}  Outputs 0\n (literal newline) for truthy, and the empty string for falsy. The idea is to subtract 1 then 2 then 3 all the way up to the input. If you hit 0, then you know this is a triangular number, so you can stop there. # Push -input on both stacks. One is a counter and the other is a running total (([({})])<>) # Count up from -input to 0 { # Push the new total which is: (counter += 1) + total (popped) + input (not popped) # This effectively adds 1, then 2, then 3 and so on to the running total (({}())<>{}({})) # If not 0 { # Push to 0s and switch stacks to "protect" the other values ((<>)) # End if } # Pop the two 0s, or empty the stack if we hit 0 {}{} # End loop }  ### Here's a 46 byte solution that I found interesting. {<>(({}())){({}[()]<>{(<({}[()])>)}{}<>)}{}<>}  Outputs 0\n (literal newline) for truthy, the empty string for falsy. The idea is to count down from input by consecutive numbers, 1 at a time. E.g. input - (1) - (1,1) - (1,1,1). Each time we subtract, if we aren't at 0 yet, we leave an extra value on the stack. That way, if we are at 0 and are still subtracting when we pop we remove the last value on the stack. If the input was a triangular number, we will end exactly at 0, and wont pop the 0. # Implicit input (call it I) # Until we reach 0, or the stack is empty { # Add 1 to the other stack and push it twice. This is our counter. <>(({}())) # While counter != 0 { # counter -= 1 ({}[()] # if I != 0 <>{ # I -= 1, and push 0 to escape the if (<({}[()])>) # End if } # Pop from the stack with I. This is either the 0 from the if, or I {} # Get ready for next loop End while <>) # End While } # Pop the counter that we were subtracting from {}<> # End Until we reach 0, or the stack is empty. }  6 # Jelly, 5 bytes ×8‘Æ²  Try it online! ### Background Let n be the input. If n is the kth triangular number, we have  $n=\frac{k\left(k+1\right)}{2}\phantom{\rule{thickmathspace}{0ex}}⟺\phantom{\rule{thickmathspace}{0ex}}{k}^{2}+k-2n=0\phantom{\rule{thickmathspace}{0ex}}⟺\phantom{\rule{thickmathspace}{0ex}}k=\frac{1}{2}\left(-1±\sqrt{1+8n}\right),$ which means there will be a natural solution if and only if 1 + 8n is an odd, perfect square. Clearly, checking the parity of 1 + 8n is not required. ### How it works ×8‘Æ² Main link. Argument: n ×8 Yield 8n. ‘ Increment, yielding 8n + 1. Æ² Test if the result is a perfect square.  5 # PowerShell, 31 30 bytes "$args"-in(1..1e6|%{($s+=$_)})

Try it online!

Nice and slow brute force method. Make an array of every sum of 1 through 106, and see if the argument is in there.

5

# Brain-Flak, 42 bytes

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


Try it online!

## Explanation

The goal of this program is to create a state on two stacks and perform constant operation on both stacks until one of them zeros, we can then output depending on which stack we are on. This is similar to programs that determine the sign of a number. These programs put n on one stack and -n on the other and add one and switch stacks until one of the stacks is zero. If the number was negative in the first place the first stack will hit zero, if the number was positive the other stack will hit zero.

Here we create two stacks one that subtracts consecutive numbers from the input and one that just subtracts one. The one that subtracts consecutive numbers will only terminate if the number is triangular, (other wise it will just pass zero and keep going into the negatives). The other one will always terminate for any positive number, but will always do so slower than the first, thus non-triangular numbers will terminate on that stack.

So how do we set up stacks so that the same operation subtracts consecutive numbers on one and subtracts one on the other? On each stack we have the input on top so that in can be checked, below that we have the difference and below that we have the difference of the difference. Each time we run we add the "difference of the difference" to the regular "difference" and subtract that from the input. For the stack that checks for triangularity we set our double difference to be 1 so that we get consecutive integers each time we run, for the other stack we set it to 0 so that we never change the difference, that is it always stays 1. Here is how the stack is set up at the beginning, where n is the input:

-n  -n
0   1
1   0


When we finally do terminate we can use these differences to check which stack we are on we pop the top two values and we get 1 for a triangular number and 0 for a non-triangular number.

### Annotated code

(([{}](<((())<>)>))<>) Set up the stack
{                      While
<>                    Switch stacks
({}({}({})))          Add bottom to second to bottom, add second to bottom to top
}                      End while
{}{}                   Pop the top two values


Here's a 50 byte solution I like as well.

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


Try it online!

5

# Cubix, 23 2425 bytes

I1Wq/)s.;0..s;p-?\.+O@u


0 for truthy and nothing 0 for falsey. Brutes forces by incrementing counter, adding to cumulative sum and comparing to input. Now to try and fit it on a 2x2x2 cube. Did it!

    I 1
W q
/ ) s . ; 0 . .
s ; p - ? \ . +
O @
u .


Try it online!

• / Reflect to to face.
• I10\ get integer input, push 1 (counter), push 0 (sum) and reflect
• +s;p- loop body. Add sum and counter, drop previous sum, raise input and subtract
• ? Test the result of the subtraction
• For 0 result carrying on straight ahead \.uO@ reflect to bottom face, no-op, U-turn, output and halt.
• For positive result turn right onto bottom face and @ halt
• For negative result turn left ;qWs)/su drop subtraction, put input to bottom, shift left, swap counter and sum, increment counter, reflect, swap sum and counter, U-turn onto main loop body.

So tantalizingly close... that last byte is going to take a lot of effort and cleverness though.
ETHproductions

Yep, thought I had it but is being elusive
MickyT

1
@ETHproductions found the byte
MickyT

Your code and your unfolded cube appear to be different, the lower right corner is a . on the cube but a 1 in your code.
Wheat Wizard

@WheatWizard Thanks for that, bad editing on my part
MickyT

4

# 05AB1E, 7 6 bytes

EDIT: Thanks to @Dennis: Saved a byte because I forgot about the increment operator

8*>t.ï


Try it online!

n is triangular if sqrt(8n + 1) is an integer

How it works

8* # multiply implicit input by 8
t # sqrt
.ï # is integer


Probably wasn't available yet at the time, but t.ï can be Å² these days, which is a builtin to check if a number is a square.
Kevin Cruijssen

4

# Perl 6, 17 bytes

{$_∈[\+] 1..$_}


Just checks whether $_, the input to the function, is equal to any of the elements of the triangular addition reduction (1, 1+2, ..., 1+2+...+$_).

4

# Alice, 38 22 bytes

A lot of bytes saved thanks to Martin and Leo

/ i \2*.2RE.h*-n/ o @



There is a trailing newline. Outputs 1 for triangular, 0 otherwise.

Try it online!

### Explanation

This uses the same approach as my CJam answer, only clumsier. In linearized form, the program becomes

i2*.2RE.h*-no@


where the i and o are actually in ordinal mode.

Consider input 21 as an example.

i         Input integer                       STACK: 21
2*        Multiply by 2                       STACK: 42
.         Duplicate                           STACK: 42, 42
2RE       Integer square root                 STACK: 42, 6
.         Duplicate                           STACK: 42, 6, 6
h         Increment                           STACK: 42, 6, 7
*         Multiply                            STACK: 42, 42
-         Subtract                            STACK: 0
n         Logical negation                    STACK: 1
o         Output integer                      STACK:
@         End program


Luis Mendo

1
I have a feeling this could be roughly halved with one of Martin's fancy control structures...
ETHproductions

So do I ... :-)
Luis Mendo

My first Alice golf: Same code, 23 bytes
Nitrodon

A more "standard" layout for this kind of program would be this. That said, you could get rid of the 1 on the stack, and simply output the logical negation of the subtraction (i.e. ...h*-no@)
Leo

4

# Japt, 10 7 bytes

Saved 3 bytes thanks to @Luke and @ETHproductions

*8Ä ¬v1


Try it online!

### Explanation:

*8Ä ¬v1
¬    // Square root of:
*8       //   Input * 8
Ä      //   +1
v1  // Return 1 if divisible by 1; Else, return 0


õ å+ øU


### Explanation:

õ å+ øU
õ           // Create a range from [1...Input]
å+        // Cumulative reduce by addition
øU     // Does it contain the input?


Try it online!

The question asks for two constant distinct ouputs.
xnor

*8Ä ¬u1 c for 9B (outputs 0 if input is triangular, 1 otherwise)
Luke

@Luke You could change u1 c to v1, I believe (switching the outputs)
ETHproductions

7 bytes? Nice! Somehow missed this while posting my own, similar solution late last might. Let me know if you'd like me to delete it.
Shaggy

4

# R, 23 19 bytes

Similar approach as other answers. Checks to see if 8x+1 is a perfect square.
-4 bytes thanks Giuseppe and MickyT.

!(8*scan()+1)^.5%%1

Try it online!

2
you can use ! instead of ==0
Giuseppe

This is extra nice since it's vectorized, too!
Giuseppe

1
I think you can get rid of the exterior brackets as well !(8*scan()+1)^.5%%1
MickyT

3

# MATL, 5 bytes

t:Ysm


Try it online!

Explanation:

t       % Duplicate input
:      % Range(1, input)
Ys    % Cumulative sum. This will push the first *n* triangular numbers
m   % ismember. Pushes true if the input is contained within the array we just pushed


I was going to post t:Ys=a. Forgot about m :-)
Luis Mendo

1
@LuisMendo I didn't know about m until I saw this answer. Funny how the two answer are almost identical :D
DJMcMayhem

3

## Batch, 72 bytes

@set/aj=i=0
:l
@if %1% gtr %j% set/aj+=i+=1&goto l
@if %1==%j% echo 1


Outputs 1 on success, nothing on failure. Works for zero too, although not requested by the question for some reason.

3

# JavaScript (ES7), 19 18 bytes

From my answer to a related question.

Outputs false for triangular numbers or true for non-triangular, as permitted by the OP.

n=>(8*n+1)**.5%1>0


## Try It

f=
n=>(8*n+1)**.5%1>0
oninput=_=>o.innerText=f(+i.value)
<input id=i type=number><pre id=o>

I think you could save a byte with n=>(8*n+1)**.5%1>0 (which would reverse the outputs)
ETHproductions

@ETHproductions: OK, as long as you're allowing it. Is doing so normally permitted, though?
Shaggy

1
It qualifies as "two constant, distinct outputs", so yes. Other decision-problem challenges may require truthy/falsy though.
ETHproductions

3

# PHP, 30 Bytes

Prints 1 for true and nothing for false

<?=fmod(sqrt(8*$argn+1),2)==1;  Try it online! fmod # PHP, 37 Bytes Prints 1 for true and nothing for false <?=($x=sqrt($q=2*$argn)^0)*$x+$x==\$q;


Try it online!

3

## Mathematica, 28 bytes

!Accumulate@Range@#~FreeQ~#&


I recommend replacing 7! by #. First, it's shorter; more importantly, the current solution is not correct, as it artificially imposes a limit on the size of the input it works on.
Greg Martin

1
OP says: "You may assume that the input is a positive integer under 10^6".But I like your idea and I will take it ,although mine gives the right result for every case using a list of 5040 elements but yours worst case needs a list of 999999 elements.Thanks for the tip!
J42161217

1
Oops sorry, didn't see the OP's comment! Yes, there are some "perverse" incentives in code golfing: that 1-byte savings is more important in a code-golf question than all the efficiency in the world :)
Greg Martin

3

# Pari/GP, 18 bytes

n->issquare(8*n+1)


Try it online!

There is a built-in to test if a number is a polygonal number, but it is one byte longer.

# Pari/GP, 19 bytes

n->ispolygonal(n,3)


Try it online!

3

# Excel, 31 22 bytes

9 bytes saved thanks to Octopus

Outputs TRUE for triangular numbers. Else FALSE. Checks if 8*n+1 is a perfect square.

=MOD(SQRT(8*B1+1),1)=0


1
=MOD(SQRT(8*A1+1),1)=0 saves a few bytes
Octopus

2

# Brachylog, 5 bytes

≥ℕ⟦+?


Try it online!

## Explanation

≥ℕ⟦+?
≥ℕ     There is a number from 0 to {the input} inclusive
⟦    such that the range from 0 to that number
+   has a sum
?  that equals the input


2

# Python - 52 bytes

Note: I know that the other two Python answers are much shorter, but this is the old-school way, more of a by-hand algorithm

n=input();i=s=0
while s<n:s=(i*i+i)/2;i+=1
print s>n


2

# APL (Dyalog), 6 bytes

⊢∊+\∘⍳

Try it online!

### Explanation

⊢∊+\∘⍳
⍳                 Creates a range from 1 to the right_argument
+\                    Cumulative sum of this range; 1 1+2 1+2+3 .. 1+2+..+n. These are the triangular numbers
⊢∊                      Does the right argument belong to this list of integers in this cumulative sum

Outputs 0 for false and 1 for true.

2

# TI-BASIC, 10 7 bytes

-3 thanks to @lirtosiast

:not(fPart(√(8Ans+1


Takes input on X. Checks if √(8X+1) is a whole number

Why not not(fPart(√(8Ans+1?
lirtosiast

@lirtosiast That'll work, thanks!
Scott Milner
আমাদের সাইট ব্যবহার করে, আপনি স্বীকার করেছেন যে আপনি আমাদের কুকি নীতি এবং গোপনীয়তা নীতিটি পড়েছেন এবং বুঝতে পেরেছেন ।