List the elements having variable valency 1 and 3?

Hint: In chemistry, an element's valence or valency is a measure of its ability to combine with other atoms to form chemical compounds or molecules. The valency of elements in the same group in the periodic table is the same. The number of electrons in an element's outer shell determines its valency.

Complete answer:
In chemistry, an element's valence or valency is a measure of its ability to combine with other atoms to form chemical compounds or molecules. The valency of elements in the same group in the periodic table is the same. The number of electrons in an element's outer shell determines its valency.
Thallium
The chemical element thallium has the symbol Tl and the atomic number 81. It's a grey post-transition metal that doesn't occur naturally. When alone, thallium looks like tin, but when exposed to sunlight, it discolours.
Thallium is prone to oxidation states of +3 and +1. The +3 state is similar to that of the other group 13 elements (boron, aluminium, gallium, indium). The +1 condition, which is much more prominent in thallium than in the elements above it, recalls the chemistry of alkali metals, and thallium(I) ions are present geologically mainly in potassium-based ores, and are treated by ion pumps in living cells in several ways similar to potassium ions ($K^+$).
The electron configuration of thallium is \[[Xe]6{s^2}4{f^{14}}5{d^{10}}6p\]
The 6p electron is simple to detach and form Tl⁺
The electrons in the 6s subshell are the easiest to detach.
Removal of these electrons gives $Tl^{3+}$ with electron configuration \[[Xe]4{f^{14}}5{d^{10}}\]
Gold
Gold is a chemical element with the symbol Au (from Latin: aurum) and atomic number 79, which makes it one of the higher atomic number elements found in nature. It is a shiny, slightly reddish yellow, thick, smooth, malleable, and ductile metal in its purest form. Gold is a transition metal and a member of group 11 of the periodic table. It's one of the least volatile chemical elements, because it's stable at room temperature. In rocks, veins, and alluvial deposits, gold is often found in its free elemental (native) shape, as nuggets or grains.
The electron configuration of gold is \[[Xe]6s4{f^{14}}5{d^{10}}\]
The 6s electron is simple to detach and form $Au^+$
The electrons in the 5d subshell are the easiest to detach.
Removal of these electrons gives $Au^{3+}$ with electron configuration \[[Xe]4{f^{14}}5{d^8}\]

Note:
In chemistry, an element's valence or valency is a measure of its ability to combine with other atoms to form chemical compounds or molecules. The valency of elements in the same group in the periodic table is the same. The number of electrons in an element's outer shell determines its valency.