Let's break down the problem step by step, using algebra to represent the unknowns.

Variables:

• Let $A$, $B$, $C$, $D$, and $E$ represent the number of cards held by players A, B, C, D, and E respectively.

Given conditions:

1. A's statement:

   • "If you (B) give me 3 cards, you will have as many as E has." B - 3 = E \quad \Rightarrow \quad B = E + 3 \tag{1}
   • "If I (A) give you 3 cards, you will have as many as D has." B + 3 = D \quad \Rightarrow \quad B = D - 3 \tag{2}

2. Combined total of A and B vs D and E:

   • "A and B together have 10 cards more than D and E together." A + B = D + E + 10 \tag{3}

3. Relation between B and C:

   • "B has 2 cards more than C." B = C + 2 \tag{4}

4. Total number of cards:

   • "The total number of cards is 133." A + B + C + D + E = 133 \tag{5}

Solving the system of equations:

From equation $(1)$, we have $B = E + 3$.
From equation $(2)$, we have $B = D - 3$.

Substitute these into equation $(3)$: $A + B = D + E + 10$ Since $B = E + 3$, we have: $A + (E + 3) = D + E + 10$ Simplifying: $A + E + 3 = D + E + 10$ A = D + 7 \tag{6}

Now we have:

• $A = D + 7$ (from equation (6)),
• $B = E + 3$ (from equation (1)),
• $B = D - 3$ (from equation (2)),
• $B = C + 2$ (from equation (4)).

Let's substitute these into equation $(5)$ (the total number of cards): $A + B + C + D + E = 133$ Using the relations for $A$, $B$, and $C$: $(D + 7) + B + (B - 2) + D + (B - 3) = 133$ Simplifying this: $(D + 7) + B + (B - 2) + D + (B - 3) = 133$ $2D + 7 + 3B - 5 = 133$ $2D + 3B + 2 = 133$ 2D + 3B = 131 \tag{7}

Now, substitute $B = D - 3$ (from equation (2)) into equation $(7)$: $2D + 3(D - 3) = 131$ $2D + 3D - 9 = 131$ $5D - 9 = 131$ $5D = 140$ $D = 28$

Now that we have $D = 28$, substitute this back to find $B$: $B = D - 3 = 28 - 3 = 25$

Thus, B has 25 cards.

Would you like further clarification on this solution? Here are some additional questions that can extend your understanding:

1. How many cards does player A have?
2. How many cards does player C have?
3. How many cards does player E have?
4. Can you verify the total number of cards by adding up all individual counts?
5. How would the problem change if B had one card more than C instead of two?

Tip: When solving a system of equations, substituting early equations into later ones helps reduce the number of variables and simplifies the solution process.