Iff
Alison has been playing with numbers again. She started by choosing a triangular number, multiplied it by 8, and added 1. She noticed something interesting about her results...
Try a few examples. Can you make a conjecture?
Once you've made a conjecture of your own, click below to see what Alison noticed:
"If $T$ is a triangular number, $8T+1$ is a square number."
Can you prove the conjecture?
You might like to have a look at this Scrambled Proof and see if you can rearrange it into the original order.
Claire thought that she could use a picture to prove this conjecture. Can you use her picture to create another proof to show that the conjecture is true?
I wonder if there are any integers $k$ where $8k+1$ is a square number but $k$ is not a triangular number...
Can you prove that if $8k+1$ is a square number, $k$ must be a triangular number?
Here is a Scrambled Proof for this conjecture.
Can you use your theorem to devise a quick way to check whether the following numbers are triangular numbers?
- 6214
- 3655
- 7626
- 8656
If $T$ is the $n^{th}$ triangular number, how could you express $T$ in terms of $n$?
What happens if you multiply that expression by $8$ and add $1$?
If you're finding it hard to prove the conjecture, you might like to print out these proof sorter cards, and then cut them out and rearrange them to form a proof. Alternatively, you can use this interactive proof sorter.
For the second conjecture, if you're finding it hard to prove, here is another set of proof sorter cards for you to print out and rearrange, and another interactive proof sorter.
Navjot from Sherborne Qatar sent us a diagram showing why the triangular numbers can be expressed using the formula $T_n=\frac12 n(n+1)$:
Rishika from Nonsuch High and Navjot both showed how to calculate the quadratic sequence by using differences. Here is Rishika's method:
Triangular number sequence: 1 3 6 10”¦
I noticed that this was a quadratic sequence, with the first differences
being 1,2,3,4.. and the second common difference being 1.
1 3 6 10 a+b+c
2 3 4 3a+b
1 1 2a
I then used the above expressions to solve and obtain the quadratic sequence rule:
$2a=1 \Rightarrow a=0.5$
$3(0.5) +b = 2$
$1.5 +b = 2$
$b = 0.5$
$a+b+c = 1$
$0.5+0.5+c = 1$
$c = 0$
Therefore, the $k^{th}$ term of the triangular number sequence is $0.5k^2 +0.5k$
Rishika, Navjot, Emily from South Hunsley and Samuel from West Bridgnorth all showed that if you multiply this by 8 and add 1, you always get a square number. Here's how Samuel worked it out:
$T_n = \frac{n^2+n}2$
$8T_n = 4n^2+4n$
$8T_n + 1 = 4n^2+4n+1$
This factorises to give $(2n+1)^2$
Eleanor from North Yorkshire gave a clear explanation for the second part of the problem:
Conjecture: If $8n+1 = m^2$ then $n$ is triangular.
Proof:
$8n+1 = 2(4n)+1$ so $8n+1$ must be odd as it is in the form $2(n)+1$
So if $8n+1$ is square then $8n+1 = (2k+1)^2$ since all odd squares have odd
roots
Then $8n+1 = 4k^2+4k+ 1$
So
$$\eqalign{n &= \frac{4k^2+4k}8 \\ &=\frac12(k^2+k) \\ &= \frac12k(k+1) \\ &= \sum_{r=1}^k r}$$
Hence if $8n+1 = m^2$ then $n$ is triangular.
Eleanor went on to show how to quickly check which numbers were triangular, by taking the square root of $8x+1$ for each value of $x$:
$\sqrt{8\times6214+1} \approx 222.9641227$ so $6214$ is not triangular.
$\sqrt{8\times3655+1} = 171$ so $3655$ is triangular.
$\sqrt{8\times7626+1} = 247$ so $7626$ is triangular.
$\sqrt{8\times8656+1} \approx 263.1520473$ so $8656$ is not triangular.
Pablo from King's College Alicante sent a diagram to show a geometric way of explaining what happens.
Pablo's Solution
Triangular numbers can be represented as a right angled triangle. For the
third triangular number the visual representation would have a base
of three, with two on top of it and one at the top (similar to the triangle in
Figure 2).
If 8 copies of one of these triangles are made, they can be arranged as
seen in Figures 1 and 2 to make an incomplete square. The only thing
missing would be an extra dot. Thus, if you take 8 copies of a Triangular
Number, plus an extra dot, a square can be made.
We also received solutions from Kristian from Maidstone Grammar, and Amrit from Hymers College. Thanks to you all!
Why do this problem?
This problem gives an opportunity to try something out, notice a pattern, make a conjecture, and then prove it. There are several approaches to the proof, both algebraic and visual, which can lead to fruitful discussion about different methods of proof.
Proving the statements in both directions (if and only if) will be quite challenging for some students. To help them to develop these skills, we have provided two "proof sorter" activities which can be used to scaffold the task and offer the support that students need in order to complete the problem.
Possible approach
If students are unfamiliar with triangular numbers, Picturing Triangle Numbers is a good starting point.
"Choose a triangular number, multiply it by 8 and add 1. What did you get?"
Collect a variety of answers on the board, and invite students to comment on anything they notice:
"The answer is always odd."
"The answer is always a square number."
Once they have made a conjecture, there are different routes to a proof. Some students may prefer a visual approach, looking for ways to show the triangular numbers fitting together to make a square. Others may prefer to work algebraically.
Once students have had a go at proving their conjecture, you may wish to offer them this Proof Sorter, which has the steps of a possible proof mixed up for them to put in the correct order. There is a second Proof Sorter which proves the result that
if $8k+1$ is square, $k$ is triangular. This could lead to a discussion about the difference between "If" and "If and only if", which can be explored further in the problem Iffy Logic.
Here are links to interactive proof sorters of the first conjecture and of the converse.
Key questions
What do you notice when you work out $8T+1$ for a triangular number $T$?
If $8k+1$ is square, does that necessarily mean $k$ is triangular?
Possible support
The problem Picturing Triangular Numbers is a good introduction to visual proof, and can be used as a foundation to a pictorial proof of this result.
Possible extension
Can you prove it? and Sums, Squares and Substantiation has a selection of similar results for which students could try to construct proofs.