Question 52

Let the large circles have radius $r$ cm.

By symmetry, AÔB $= \tfrac{360^{\circ}}{4}=90^{\circ}$

$\therefore$ by Pythagoras’ theorem,

So, the large circles have radius $1+\sqrt{2}$ cm.

Since $7!=5040$, none of the digits can be $7$, $8$, or $9$, since the sum of the factorials must have $3$ digits.

For this reason, the value of the three-digit number must be less than $700$. This means that none of the digits can be $6$, since $6!=720$.

Since the digits are distinct, the largest value of the sum of the factorials is $5!+4!+3!=150$. So, the first digit must be $1$.

Also, one of the remaining digits must be a $5$, otherwise the sum of the factorials will have less than $3$ digits.

The remaining unknown digit must therefore be one of $2$, $3$, or $4$. By checking each possibility, we find that the number is $145=1!+4!+5!$ .