Suppose we have a smart contract that is able to exchange CT tokens for TT tokens and TT tokens for CT tokens. It is based on a constant product formula. The number of CT tokens times the number of TT tokens is always equal to 200.

The current market price of a CT token is 5 ETH and the current market price of a TT token is 10 ETH.

To allow swaps, the contract is initially provide with 20 CT tokens and 10 TT tokens.

Note: This is a constant product AMM.

If a trader sends 5 CT tokens to the contract, how many TT tokens should she expect in exchange? (We are not concerned with fees.)

Question

Suppose we have a smart contract that is able to exchange CT tokens for TT tokens and TT tokens for CT tokens. It is based on a constant product formula. The number of CT tokens times the number of TT tokens is always equal to 200.

The current market price of a CT token is 5 ETH and the current market price of a TT token is 10 ETH.

To allow swaps, the contract is initially provide with 20 CT tokens and 10 TT tokens.

Note: This is a constant product AMM.

If a trader sends 5 CT tokens to the contract, how many TT tokens should she expect in exchange? (We are not concerned with fees.)

BlackTom AI · Accepted Answer

Start by identifying the constant product invariant: CT_tokens_in_pool × TT_tokens_in_pool = 200, which is the k value.
Initially, the pool has 20 CT and 10 TT, and 20 × 10 = 200, so the invariant holds.
Now a trader sends 5 CT tokens to the contract. The CT side of the pool increases to 20 + 5 = 25. To keep the product equal to 200, the TT side must satisfy 25 × TT_final = 200, so TT_final = 200 / 25 = 8.
The trader thus removes TT tokens from the pool: TT_given = TT_initial − TT_final = 10 − 8 = 2.
Proceeding option by option:
- Option 1 (1): If 1 TT were sent out, the final TT in the pool would be 9, yielding 25 × 9 = 225, which violates the invariant; thus this option cannot occur.
- Option 4 (4): If 4 TT were sent out, the final TT would be 6, giving 25 × 6 = 150, again not equal to 200; this is inconsistent with the invariant.
- Option 2 (2): This matches the required math: 25 CT in, 8 TT left in the pool, so 10 − 8 = 2 TT to the trader, and 25 × 8 = 200 preserves the invariant.
- Option 3 (3): If 3 TT were sent out, the final TT would be 7, producing 25 × 7 = 175, which does not maintain the constant product; thus it is incorrect.
- Option 5 (5): If 5 TT were sent out, the final TT would be 5, producing 25 × 5 = 125, far from 200; this also fails the invariant.
In summary, only the scenario yielding 2 TT to the trader preserves the constant product relationship, while the others would violate the invariant and are therefore incorrect.

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