I am terrible at doing mental math, so I was recommended to play this simple game to hone my number sense. The game involves picking four numbers and the coming generating the numbers from $1$ to $20$.
Here's an example: Let the set of numbers be $\left \{ 2,3,4,5 \right \}$.
The numbers generated are as follows:
$1 = \left ( 3-2 \right )\times \left ( 5-4 \right )$
$2 = \left ( 3-2 \right )+ \left ( 5-4 \right )$
$3 = 3 \times \left ( \left ( 4+2 \right ) \times 5\right ))$
$4 = \left ( 4-2 \right )+ \left ( 5-3 \right )$
$5 = 2 + \frac{4+5}{3}$
$6 = 3 + \frac{4+5}{2}$
$7 = \left ( 2 \times 3 \right )+ \left ( 5-4 \right )$
$8 = \left ( 2+4 \right )+ \left ( 5-3 \right )$
$9 = \left ( 4+5 \right )+ \left ( 3-2 \right )$
$10 = \left ( 2 \times 5 \right )\times \left ( 4-3 \right )$
$11 = \left ( 2 \times 5 \right )+ \left ( 4-3 \right )$
$12 = \left ( 2 + 4 \right )\times \left ( 5-3 \right )$
$13 = \left ( 4-2 \right )\times 5 + 3$
$14 = 2 + 3+ 4 + 5$
$15 = \left ( 2 \times 3 \right )+ \left ( 4+5 \right )$
$16 = 4 \times \left [ \left ( 5-3 \right ) +2 \right ]$
$17 = \left ( 3 \times 5 \right )+ \left ( 4 - 2 \right )$
$18 = 3^{2} + 4 + 5$
$19 = \left ( 4 \times 5 \right )- 3 + 2$
$20 = \left ( 4 \times 5 \right )\times \left ( 3-2 \right )$
However this game becomes difficult with a different set of numbers consisting of primes $\left \{ 2,3,5,7 \right \}$. The number $5$ couldn't be obtained.
So does this type of game actually help in getting good at mental math?
I used to play this with my daughters, only the game was to create the number 24. (There is a commercially available set of cards with doable problems; the game is to be the first to get 24 on each card, and collect as many wins as possible.
It did help them get good at elementary school math.
By the way, $$ (2^3 -7) \times 5 = 5 $$