The eccentric privateer Sir Francis Duck disliked being interrupted while playing bowls, but the situation was urgent.
"Cap'n Duck?" said the ship's mate nervously, tapping him on the shoulder.
"What is it?" asked Duck, irritably.
"The King of Spain's fleet is approaching the coast of Plymouth, bent on invasion!"
"Then we shall defend the realm," Duck replied calmly.
"With what? We're out of cannonballs."
"We will use bowls instead," said the ever-inventive Duck, pointing to a regular-looking pile of perfectly round bowls.
"How many are there in that pile, Cap'n?"
"At this club," said Duck, "each pile of bowls can be arranged in two distinct patterns. At the moment they form a pyramid. At the base is a triangle, whose rows are successive whole numbers, so the top triangle contains 1 bowl, the next 1+2=3, the next 1+2+3=6, and so on."
"Those layers are triangular numbers, aren't they Cap'n? Formed by multiplying two consecutive integers together and halving the result? Like (4x5)÷2=10?"
"I'm glad you remember your seaman's lore. So the entire pile contains a sum of successive triangular numbers—"
"Known as a tetrahedral number. Take three consecutive integers, multiply them, and divide by 6. Like (3x4x5)÷6=10."
"Good. Now, notice that the number 10 is both a tetrahedral number, being 1+3+6, and a triangular number, being 1+2+3+4. And, indeed, that pile comprises a number that is both triangular and tetrahedral."
"There's more than ten in that pile, right?"
"There are 120, which, being (15x16)÷2 is the 15th triangular number, and being (8x9x10)÷6 is the 8th tetrahedral number."
"120 cannonballs won't stop the fleet!"
"No. We need the next largest arrangement that is triangular and tetrahedral.
What number does Sir Francis need?
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The answer
The next number that is both tetrahedral and triangular is 1,540. It is the 55th triangular number, (55x56)÷2, and the 20th tetrahedral number, (20x21x22)÷6. The simplest way to establish this is to calculate tetrahedral numbers in turn. Amazingly, there is a larger number with this property: 7,140. It is the 119th triangular number, (119x120)÷2, and the 34th tetrahedral number, (34x35x36)÷6. It is also the largest such number, so the complete list is 1, 10, 120, 1,540, and 7,140. This was proved by ET Avanesov in 1966.
The winner was Charles Goldie from Brighton
