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Siobhan, Ruth, Ruth and Toby from Risley Lower Grammar Primary School, Joe from Hove Park Lower School, Nathan, Dulan and Pavan from Wilson's Grammar School and Thomas from PS6 in New York noticed that:

If you double a triangular number, you get a rectangular number.

For the $n$th triangle number, the sides of the rectangle are $n$ and $n+1$.

Therefore, the $n$th triangular number can be written $n(n+1)/2$.

Using this $T_{250} = 31375$

Students from the Tower Hamlets Enriching Maths project also worked on this problem:

We worked out that the rectangle always has a length which is 1 more than the width.

So to find the nth triangle number, work out n times (n+1) and then divide by 2.

To tell if 4851 is a triangle number, we doubled it to get 9702, and then used trial and improvement to see if we could find two consecutive numbers which multiplied to give 9702.

98 and 99 work, so 4851 is the 98th triangle number.

Elise from Trentham High School sent us this complete solution to the problem.

Jinquan from The Chinese High School in Singapore explained
that:

$T_{250}+T_{250}$ is $250 \times251$, and more generally $T_{n}+T_n
= n(n+1)$

Hence, $T_n=n(n+1)/2$

which gives $T_{250}=31375$,

Consider $4851$.

If $4851$ is a triangular number, $9702$ can be expressed as $n(n+1)$.

By solving the quadratic equation or by estimating, we get $n = 98$, and hence $4851 = T_{98}$

In general, a number $x$ is a triangular number if and only if $n(n+1)=2x$ is solvable for positive integers of $n$.

Consider $6214$.

If it is a triangular number, $12428$ can be expressed as $n(n+1)$.

By solving the quadratic equation or by estimating, we see that there are no solutions in positive integers.

Hence $6214$ is not a triangular number.

Consider $3655$.

If it is a triangular number, $7310$ can be expressed as $n(n+1)$.

By solving the quadratic equation or by estimating, we get $n = 85$, and hence $3655 = T_{85}$

Using similar thinking leads to the conclusion that $7626$ is
$T_{123}$ and that $8656$ is not a triangular number.

Well done to you all.