Matt Parker mistake in cannonball stacking video

Question

I could need someone to do a check on this problem, since it has been published in Matt Parkers book "things to make and do in the fourth dimension" and has now been featured in a numberphile video

In Matt Parker's most recent video on numberphile he states that you can build a giant hexagon with sides of length 22 and it will take 946 balls. It looks like he has used these numbers when he in reality should have been using these so he got the layers to be $1,6,15,\ldots,946$ instead of $1,7,19,\ldots,1387$.

The problem he was trying to solve was if there exists a number of cannonballs where they can both form a flat hexagon and a hexagonal pyramid (inspired by this old problem).

I made a simple python programme to check for a solution of the height up to $10^7$ (which equals $10^{18}$ cannonballs) with no solution

layer_size = 1
sum_to_height = 1
totals = []
layers = []
for height in range(2,10000):
    layer_size = 3*((height-1)**2+height-1)+1 #Formula for how many balls in each layer
    sum_to_height += layer_size   #Sum of all balls in pyramid to current height
    totals.append(sum_to_height)
    layers.append(layer_size)

print(set(layers).intersection(totals)) #If there is a intersect, then that is a solution

I believe he is using the wrong sequence based off the problem he is trying to solve — Gravrok, Apr 10 '19 at 18:50

score 4 · Answer 1 · answered Apr 10 '19 at 19:08

If you look at the video it is clear that he really meant hexagonal numbers OEIS A000384, where $A000384(22) = 946$ and the sum of the first $11$ of these is also $946$. You are using a closely related but different sequence OEIS A003215 which is centered hexagonal numbers. The hexagonal array of balls look very superficially the same, but only the centered version has a center with six triangles radiating outwards. The version used in the video has four triangles radiating from a single vertex of the hexagon.

Matt Parker mistake in cannonball stacking video

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