Problem statement: Each figure shows four vertex numbers and a label $S3$ which equals three times the sum of the four vertex numbers. Determine the missing vertex numbers or the missing $S3$ values for the eight figures. 1. Figure 1: vertices $2,1,2,3$ and $S3=24$. Compute the sum: $2+1+2+3=8$. Check $S3=3(8)=24$, so the figure is consistent and there are no missing values. 2. Figure 2: vertices $4,3,0,6$ and $S3=39$. Compute the sum: $4+3+0+6=13$. Check $S3=3(13)=39$, so the figure is consistent and there are no missing values. 3. Figure 3: vertices $1,2,4,4$ and $S3=33$. Compute the sum: $1+2+4+4=11$. Check $S3=3(11)=33$, so the figure is consistent and there are no missing values. 4. Figure 4: vertices $5,1,0,2$ and $S3$ is blank. Compute the sum: $5+1+0+2=8$. Therefore $S3=3(8)=24$. So the missing $S3$ is $24$. 5. Figure 5: vertices $x,2,y,6$ and $S3=45$. We have $S3=3(x+2+y+6)=45$. Divide by 3 to get the vertex sum: $x+2+y+6=15$. So $x+y=7$. There are infinitely many solutions; integer nonnegative examples are $(x,y)=(0,7),(1,6),(2,5),(3,4),(4,3),(5,2),(6,1),(7,0)$. 6. Figure 6: vertices $5,3,0,z$ and $S3=30$. Compute required sum: $S3/3=30/3=10$. Known sum of three given vertices: $5+3+0=8$. So the missing value $z=10-8=2$. 7. Figure 7: vertices $0,0,0,w$ and $S3=0$. Compute required sum: $S3/3=0/3=0$. Known sum of three zeros is $0$. So the missing value $w=0$. 8. Figure 8: vertices $3,u,v,5$ and $S3=60$. Compute required sum: $S3/3=60/3=20$. Known partial sum: $3+5=8$. So $u+v=20-8=12$. There are infinitely many solutions; integer nonnegative examples are $(u,v)=(0,12),(1,11),(2,10),(3,9),(4,8),(5,7),(6,6),(7,5),(8,4),(9,3),(10,2),(11,1),(12,0)$. Final answers summary: Figure 1: consistent ($S3=24$). Figure 2: consistent ($S3=39$). Figure 3: consistent ($S3=33$). Figure 4: missing $S3=24$. Figure 5: $x+y=7$ (examples given). Figure 6: missing vertex $=2$. Figure 7: missing vertex $=0$. Figure 8: $u+v=12$ (examples given).