Define variables for the number of students in each region of the Venn diagram.

Let $V$, $T$, and $H$ represent the sets of students who played Volleyball, Tennis, and Hockey, respectively.
Let $V_o$ be the number of students who played Volleyball only.
Let $T_o$ be the number of students who played Tennis only.
Let $H_o$ be the number of students who played Hockey only.
Let $VT_o$ be the number of students who played Volleyball and Tennis only (not Hockey).
Let $VH_o$ be the number of students who played Volleyball and Hockey only (not Tennis).
Let $TH_o$ be the number of students who played Tennis and Hockey only (not Volleyball).
Let $VTH$ be the number of students who played all three games ($|V \cap T \cap H|$). This is the value we need to find.
Let $N_0$ be the number of students who played none of the games.

We are given:

• Total students = 122
• $|V| = 59$
• $|T| = 58$
• $|H| = 53$
• $V_o = 28$
• $|V \cap T| = 22 \implies VT_o + VTH = 22$
• $|T \cap H| = 27 \implies TH_o + VTH = 27$
• $N_0 = VTH$ (Those who did not play any games were equal to those who played all three games).

From the intersection data:
$VT_o = 22 - VTH$
$TH_o = 27 - VTH$

Now use the total for each sport:
For Volleyball: $|V| = V_o + VT_o + VH_o + VTH = 59$
Substitute $V_o = 28$ and $VT_o = 22 - VTH$:
$28 + (22 - VTH) + VH_o + VTH = 59$
$50 + VH_o = 59$
$VH_o = 59 - 50 = 9$

For Hockey: $|H| = H_o + VH_o + TH_o + VTH = 53$
Substitute $VH_o = 9$ and $TH_o = 27 - VTH$:
$H_o + 9 + (27 - VTH) + VTH = 53$
$H_o + 36 = 53$
$H_o = 53 - 36 = 17$

For Tennis: $|T| = T_o + VT_o + TH_o + VTH = 58$
Substitute $VT_o = 22 - VTH$ and $TH_o = 27 - VTH$:
$T_o + (22 - VTH) + (27 - VTH) + VTH = 58$
$T_o + 49 - VTH = 58$
$T_o = 58 - 49 + VTH = 9 + VTH$

The sum of all distinct regions must equal the total number of students:
Total students = $V_o + T_o + H_o + VT_o + VH_o + TH_o + VTH + N_0$
Substitute all expressions in terms of $VTH$ and $N_0 = VTH$:

Combine the constant terms:

Combine the $VTH$ terms:

So the equation becomes:

The number of students who played all three games is $VTH = 10$. The bursary is awarded to students who can play all three games.

The coach should forward $\boxed{\text{10 students}}$ for consideration of the bursary.