Question

dominos game
the equivalent measures game uses dominoes. there is only one rule to
this game. by studying the examples below, can you figure out the rule?
what is the rule?
type here

Explanation:

Step1: Analyze First Example

Start with the first domino set. The start domino has 1 dot. Then two dominoes: first has 6 and 1 dots, second has 3 and 6 dots. The finish domino has 3 dots. Let's calculate sums: (6 + 1) = 7, (3 + 6) = 9? No, maybe count dots on each half. Wait, start is 1. Then dominoes: first domino top 6, bottom 1; second domino top 3, bottom 6. Wait, maybe group by dot count. Wait, start dot: 1. Then dominoes: 6+1=7, 3+6=9? No. Wait, finish is 3. Wait, 1 (start) + 6 (top of first domino) + 3 (top of second domino) = 10? No. Wait, maybe count the number of dominoes? No. Wait, another approach: look at the dots on each domino's halves. First example: start is 1 dot. Then two dominoes: first domino (top: 6, bottom: 1), second domino (top: 3, bottom: 6). The finish is 3 dots. Wait, 6 (top of first) + 1 (bottom of first) = 7; 3 (top of second) + 6 (bottom of second) = 9. No. Wait, start is 1. Then dominoes: 6 (top) + 3 (top) = 9; 1 (bottom) + 6 (bottom) = 7. No. Wait, second example: start is 4 dots (4 dots on start). Then four dominoes: first (top: 8, bottom: 4), second (top: 2, bottom: 8), third (top: 1, bottom: 2), fourth (top: 6, bottom: 1). Finish is 6 dots. Wait, start: 4. Dominoes: 8+4=12, 2+8=10, 1+2=3, 6+1=7. No. Wait, finish is 6. Start is 4. 4 + 8 (top of first) + 2 (top of second) + 1 (top of third) + 6 (top of fourth) = 4+8+2+1+6=21. No. Wait, maybe the number of dots on the start plus the sum of the top dots of the dominoes equals the sum of the bottom dots plus the finish? Wait, first example: start (1) + top1 (6) + top2 (3) = 1+6+3=10. Bottom1 (1) + bottom2 (6) + finish (3) = 1+6+3=10. Oh! That works. Let's check: 1 (start) + 6 (top of first domino) + 3 (top of second domino) = 10. 1 (bottom of first domino) + 6 (bottom of second domino) + 3 (finish) = 1+6+3=10. Yes! Now second example: start is 4 dots. Dominoes: first (top: 8, bottom: 4), second (top: 2, bottom: 8), third (top: 1, bottom: 2), fourth (top: 6, bottom: 1). Finish is 6. Let's calculate: start (4) + top1 (8) + top2 (2) + top3 (1) + top4 (6) = 4+8+2+1+6=21. Bottom1 (4) + bottom2 (8) + bottom3 (2) + bottom4 (1) + finish (6) = 4+8+2+1+6=21. Perfect! Third example: start is 2 dots. Dominoes: first (top: 4, bottom: 2), finish is 4 dots. Let's check: start (2) + top1 (4) = 6. Bottom1 (2) + finish (4) = 6. Yes! So the rule is: The sum of the number of dots on the start domino (or start dot count) plus the sum of the dots on the top halves of all the dominoes in the sequence equals the sum of the dots on the bottom halves of all the dominoes plus the number of dots on the finish domino. In formula: \( \text{Start Dots} + \sum (\text{Top Dots of Dominoes}) = \sum (\text{Bottom Dots of Dominoes}) + \text{Finish Dots} \)

Step2: Verify with Second Example

Start dots: 4. Dominoes: first (top: 8, bottom: 4), second (top: 2, bottom: 8), third (top: 1, bottom: 2), fourth (top: 6, bottom: 1). Sum of top dots: 8 + 2 + 1 + 6 = 17. Sum of bottom dots: 4 + 8 + 2 + 1 = 15. Finish dots: 6. Left side: 4 + 17 = 21. Right side: 15 + 6 = 21. Correct.

Step3: Verify with Third Example

Start dots: 2. Domino: first (top: 4, bottom: 2). Sum of top dots: 4. Sum of bottom dots: 2. Finish dots: 4. Left side: 2 + 4 = 6. Right side: 2 + 4 = 6. Correct.

Answer:

The rule is that the sum of the start dots and the dots on the top halves of all dominoes in the sequence equals the sum of the dots on the bottom halves of all dominoes and the finish dots (i.e., \( \text{Start Dots} + \sum (\text{Top Dots of Dominoes}) = \sum (\text{Bottom Dots of Dominoes}) + \text{Finish Dots} \)).