2-digit Square

A 2-Digit number is squared. When this 2-digit number is reversed and squared, the difference between the squares is also a square. What is the 2-digit number?

Consecutive Squares

The squares of any 8 consecutive numbers can be arranged into two sets of four numbers with the same sum. True of false?

Plus Minus

Can you explain the surprising results Jo found when she calculated the difference between square numbers?

Order the Products

Age 14 to 16 Short Challenge Level:

186$\times$214 (smallest)
210$\times$190
195$\times$205
198$\times$202
200$\times$200 (largest)

Numerically, using 200$\times$200
210$\times$190 = (200 + 10)(200 $-$ 10) = 200$^2$ + 2000 $-$ 2000 $-$ 100
= 200$^2-$ 100

195$\times$205 = (200 $-$ 5)(200 + 5) = 200$^2$ $-$ 1000 + 1000 $-$ 25
= 200$^2-$ 25

198$\times$202 = (200 + 2)(200 $-$ 2) = 200$^2-$ 4

186$\times$214 = (200 + 14)(200 $-$ 14) = 200$^2-$ 14$^2$

$\therefore$ 186$\times$214 $\lt$ 210$\times$190 $\lt$ 195$\times$205 $\lt$ 198$\times$202 $\lt$ 200$\times$200

Writing the numbers in terms of 200
All of the products are $(200 +n)(200 -n)$ for some small value of $n$.
\begin{align}(200+n)(200-n)&=200^2+200n-200n-n^2\\ &=200^2-n^2\end{align}

$n$ larger $\Rightarrow200^2-n^2$ smaller (for $n\gt0$).

$\therefore$ 186$\times$214 $\lt$ 210$\times$190 $\lt$ 195$\times$205 $\lt$ 198$\times$202 $\lt$ 200$\times$200

Diagrammatic representation
Red strip and green strip both have width $1$, area $200$

Red strip and green strip both have width $n$, area $200n$

You can find more short problems, arranged by curriculum topic, in our short problems collection.