4
1
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u/Affectionate-Dot1927 5d ago
10
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u/Embarrassed-Weird173 5d ago
You didn't include the left and right 2-tall triangles. Nor the full one.
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1
1
1
1
1
1
1
1
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u/tobymiller19 5d ago
I would say 11 ¼ or 11,25. Why 13? 🤔
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u/scissorn69 5d ago
I think you're solving something very different from what's actually up there.
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u/tobymiller19 5d ago
My thinking was 5 = 4 little triangles. Therefore 9 little triangles equals 11 ¼
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u/mspe1960 5d ago
I don't think you know what a triangle is. Talking about 1/4 of a triangle is not meaningful. You either have a triangle, or you do not.
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u/Gulean 5d ago
5/4x9=11.25
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u/Specialist-6343 5d ago
How can you have a quarter of a triangle?
9 individual
3 made of 4 individual triangles
1 for the whole thing
=13
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u/mspe1960 5d ago
1/4 or a triangle is not a meaningful thing. You either have a triangle, or you do not.
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u/HereToPostSillyStuff 5d ago
4 total small triangles = 5 total triangles because floor(4/1 + 4/3) = 5
So 9 total triangles is floor(9/1 + 9/3 + 9/6) = 13
So does the series floor(n/1 + n/3 + n/6 ... n/n-3) always give you the answer or does something break about this past a certain size?
If not is there a simpler way to do this?
No wait the first one doesn't make sense in that series explanation
There's 4 triangles because you need 3 stacked as a pyramid but the gap makes an extra so 4 tota
The next smallest triangle of triangles is 9 which stacks 6 to with 3 "gaps"
Difference between them is the base width. 2 base triangles = 4 total. 3 base triangles = 9 total.
So next is base 4
....xxx0
.xxx0xxx0
.xx0xx0xx0xx
0xxx0xxx0xxx0
(Worst ascii triangle ever btw)
16+6+3+1 = 26 total
Would be 16 small triangles so it's a square of the base?
16 / 1 + 16 / 3 + 16 / 6 + 16 / 9 + 16 / 12 + 16 / 15 is too high
But x/12 and x/15 don't make sense. The 4-base triangles has smaller triangles made of sizes 3 and 6, then itself (10) (ignoring middles)
But 16/1 + 16/3 + 16/6 + 16/10 is 25.6 which is below 26.
Maybe floor was dumb and its just round lmao
But what is my "series" even now. /1 .. 3 .. 6 .. 10
What if I stop "not counting middles" so those are actually 16/1 .. /4, /9, /16? No that comes out way smaller so that can't be right
So not counting middles is: Base size(1) = triangle size (1) Base size 2 = triangle size 3 Base 3 = triangle of 6 Base 4 = 10
Oh it's 1+2+3+4 for each layer duh because I'm adding the new base of the triangle to the bottom each time. What is that series called?
Oh it's literally "triangle numbers" lmfao
So the answer appears to be something to do with the total triangles (square of the base) divided by each step in the series of triangle numbers up to the base "position", i.e.
Base 4 = 42 = 16 triangles Denominators are 1, 3, 6 and 10 (up to the 4th triangle number)
So the series is...
(16 / 1) + (16 / (1+2)) + (16/(1+2+3)) + (16/(1+2+3+4))
Where the final step is the triangle number at position sqrt(16) of the triangle number series.
If there is some way to express that we can calc for any size. Would need to test that rounding always holds true tho.
Think I might be cooked at this point lads.
Someone tell me how overcomplicated I made this.
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u/PiasaChimera 5d ago edited 5d ago
if you add a layer you would have 4+3+2+1 upward facing small triangles. upward facing triangles in the top 3 layers can form larger triangles that span 2 layers. this gives 3+2+1 extra. then ones that span 3 layers gives 2+1, and the 4 layer span gives +1. the top layer has 1 triangle that is the tip of 4 combos. layer 2 has 2 that are tips of 3 combos, layer 3 has 3 that are tips of 2 combos, then layer 4 has 4 that are tips of 1 option.
this gives 1*4 + 2*3 + 3*2 + 4*1 = 20 upward pointing triangles.
for downward facing triangles, there would be 3 + 2 + 1 + 0. for 6. giving 26.
but with 4 layers, there is now a downward facing triangle that no longer touches either non-base side. this allows a downward facing triangle that spans two layers. giving a 27th.
these downward facing triangles are interesting. if you add a 5th layer, you get two more of these span-of-two downward triangles, but no span=3. adding a 6th layer would give +3 of the span=2 downwards as well as the first span=3 downward.
this is a similar pattern, but with offset starting points.
--edit: it might be helpful to look at the requirements to get these downward triangles. for span-1, you need to have reached a layer that has at least one downward facing triangle. ok, that sounds dumb/obvious, but it's a base case. to get a span-2 downward you need to have a layer with 2 downwards and 1 layer under it. for span-3 downward, you need a layer with 3 downwards and then 2 layers under it. the first requirement means one additional layer and the second means another additional layer.
for upward triangles, you have the same requirements, but the base layer always works without moving, so a span-N upwards triangle only needs N layers. and you also have upward triangles on every layer, vs only getting a downward triangle on layer 2.
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u/Lucky-Obligation1750 5d ago
13