To solve this problem, we can use stoichiometry. First, we need to write down the balanced chemical equation for the complete combustion of methane ($$\mathrm{CH}_4$$).

The balanced equation for combustion of methane is:

$$\mathrm{CH}_4+2 \mathrm{O}_2 \rightarrow \mathrm{CO}_2+2 \mathrm{H}_2 \mathrm{O}$$

This equation tells us that one mole of methane reacts with two moles of oxygen to produce one mole of carbon dioxide and two moles of water.

Next, we should find the molar mass of methane ($$\mathrm{CH}_4$$) using the given molar masses:

Molar mass of $$\mathrm{CH}_4$$ = molar mass of C + 4 * molar mass of H

Molar mass of $$\mathrm{CH}_4$$ = $$12.0\ \mathrm{g/mol}$$ (for C) + $$4 * 1.0\ \mathrm{g/mol}$$ (for H)

Molar mass of $$\mathrm{CH}_4$$ = $$12.0\ \mathrm{g/mol} + 4.0\ \mathrm{g/mol}$$

Molar mass of $$\mathrm{CH}_4$$ = $$16.0\ \mathrm{g/mol}$$

Now, to determine the mass of methane required to produce $$22\ \mathrm{g}$$ of $$\mathrm{CO}_2$$, we should find out how many moles of $$\mathrm{CO}_2$$ there are in $$22\ \mathrm{g}$$ and use the mole ratio from the balanced equation to find the moles of methane required.

Moles of $$\mathrm{CO}_2$$ = mass of $$\mathrm{CO}_2$$ / molar mass of $$\mathrm{CO}_2$$

Molar mass of $$\mathrm{CO}_2$$ = molar mass of C + 2 * molar mass of O

Molar mass of $$\mathrm{CO}_2$$ = $$12.0\ \mathrm{g/mol}$$ + $$2 * 16.0\ \mathrm{g/mol}$$

Molar mass of $$\mathrm{CO}_2$$ = $$12.0\ \mathrm{g/mol} + 32.0\ \mathrm{g/mol}$$

Molar mass of $$\mathrm{CO}_2$$ = $$44.0\ \mathrm{g/mol}$$

Moles of $$\mathrm{CO}_2$$ = $$\frac{22\ \mathrm{g}}{44.0\ \mathrm{g/mol}}$$

Moles of $$\mathrm{CO}_2$$ = $$0.5\ \mathrm{moles}$$

We will use the stoichiometric ratio from the balanced chemical equation to find the moles of $$\mathrm{CH}_4$$ required to produce $$0.5\ \mathrm{moles}$$ of $$\mathrm{CO}_2$$:

$$1\ \mathrm{mole\ of\ CH}_4 : 1\ \mathrm{mole\ of\ CO}_2$$

This means that we also require $$0.5\ \mathrm{moles}$$ of $$\mathrm{CH}_4$$.

Finally, we need to convert moles of methane to grams to find the mass:

Mass of $$\mathrm{CH}_4$$ = moles of $$\mathrm{CH}_4$$ * molar mass of $$\mathrm{CH}_4$$

Mass of $$\mathrm{CH}_4$$ = $$0.5\ \mathrm{moles} * 16.0\ \mathrm{g/mol}$$

Mass of $$\mathrm{CH}_4$$ = $$8.0\ \mathrm{g}$$

Therefore, $$8.0\ \mathrm{g}$$ of methane is required to produce $$22\ \mathrm{g}$$ of carbon dioxide after complete combustion.