Two cubes 3

Two cubes made of plasticine have single-digit integer edge lengths differing by 1 cm. Can a single larger cube with an integer edge length be made from the same total amount of plasticine?

Result

x=

Yes

No

Step-by-step explanation:

$$\begin{gathered} b=a-1 \\ {V}_1=a^3 \\ {V}_2=b^3 \\ {V}_3=V_1+V_2 \\ {a}_1=1\text{cm} \\ {b}_1=a_1+1=1+1=2\text{cm} \\ {V}_1=a_1^3+b_1^3=1^3+2^3=9{\text{cm}}^3 \\ {x}_1=\sqrt[3]{V_1}=\sqrt[3]{9}\doteq 2.0801\text{cm} \\ {a}_2=2\text{cm} \\ {b}_2=a_2+1=2+1=3\text{cm} \\ {V}_2=a_2^3+b_2^3=2^3+3^3=35{\text{cm}}^3 \\ {x}_2=\sqrt[3]{V_2}=\sqrt[3]{35}\doteq 3.2711\text{cm} \\ {a}_3=3\text{cm} \\ {b}_3=a_3+1=3+1=4\text{cm} \\ {V}_3=a_3^3+b_3^3=3^3+4^3=91{\text{cm}}^3 \\ {x}_3=\sqrt[3]{V_3}=\sqrt[3]{91}\doteq 4.4979\text{cm} \\ {a}_4=4\text{cm} \\ {b}_4=a_4+1=4+1=5\text{cm} \\ {V}_4=a_4^3+b_4^3=4^3+5^3=189{\text{cm}}^3 \\ {x}_4=\sqrt[3]{V_4}=\sqrt[3]{189}\doteq 5.7388\text{cm} \\ {a}_5=5\text{cm} \\ {b}_5=a_5+1=5+1=6\text{cm} \\ {V}_5=a_5^3+b_5^3=5^3+6^3=341{\text{cm}}^3 \\ {x}_5=\sqrt[3]{V_5}=\sqrt[3]{341}\doteq 6.9864\text{cm} \\ {a}_6=6\text{cm} \\ {b}_6=a_6+1=6+1=7\text{cm} \\ {V}_6=a_6^3+b_6^3=6^3+7^3=559{\text{cm}}^3 \\ {x}_6=\sqrt[3]{V_6}=\sqrt[3]{559}\doteq 8.2377\text{cm} \\ {a}_7=7\text{cm} \\ {b}_7=a_7+1=7+1=8\text{cm} \\ {V}_7=a_7^3+b_7^3=7^3+8^3=855{\text{cm}}^3 \\ {x}_7=\sqrt[3]{V_7}=\sqrt[3]{855}\doteq 9.4912\text{cm} \\ {a}_8=8\text{cm} \\ {b}_8=a_8+1=8+1=9\text{cm} \\ {V}_8=a_8^3+b_8^3=8^3+9^3=1241{\text{cm}}^3 \\ {x}_8=\sqrt[3]{V_8}=\sqrt[3]{1241}\doteq 10.7463\text{cm} \\ x=0 \end{gathered}$$

Try another example