Write down the number each child should say based on this rule for the arrangement shown below.

Question

Homi · Accepted Answer

The rule is that each child says the number of people standing in front of them who are taller than them.

Step 1 — Understanding the Rule

Let us understand what the rule means. "In front of them" means to their left in the line. We count how many children to the left are taller. This count is the number the child says.

Step 2 — First Arrangement: Calculating Numbers

Let us look at the children in the first arrangement, from left to right. We will compare their heights visually. Let us assign relative heights to make comparisons clear. Child 1 (Yellow, leftmost): Shortest. Let its height be 1. Child 2 (Blue): Taller than Child 1. Let its height be 4. Child 3 (Yellow, middle): Taller than Child 1, shorter than Child 2. Let its height be 3. Child 4 (Green): Taller than Child 1, Child 2, Child 3. Let its height be 6. Child 5 (Orange): Taller than Child 1, shorter than Child 2, Child 3, Child 4. Let its height be 2. Child 6 (Red): Tallest. Let its height be 7. Child 7 (Pink, rightmost): Taller than Child 1, Child 2, Child 3, Child 5, shorter than Child 4, Child 6. Let its height be 5.

So, the children and their relative heights are: C1 (Yellow): 1 C2 (Blue): 4 C3 (Yellow): 3 C4 (Green): 6 C5 (Orange): 2 C6 (Red): 7 C7 (Pink): 5

Now, we apply the rule for each child.

For Child 1 (Yellow, height 1): People in front: None. Taller people in front:

$\boxed{0}$

For Child 2 (Blue, height 4): People in front: Child 1 (Yellow, height 1). Is Child 1 (height 1) taller than Child 2 (height 4)? No. Taller people in front:

$\boxed{0}$

For Child 3 (Yellow, height 3): People in front: Child 1 (Yellow, height 1), Child 2 (Blue, height 4). Is Child 1 (height 1) taller than Child 3 (height 3)? No. Is Child 2 (height 4) taller than Child 3 (height 3)? Yes. Taller people in front:

$\boxed{1}$

For Child 4 (Green, height 6): People in front: Child 1 (Yellow, height 1), Child 2 (Blue, height 4), Child 3 (Yellow, height 3). Is Child 1 (height 1) taller than Child 4 (height 6)? No. Is Child 2 (height 4) taller than Child 4 (height 6)? No. Is Child 3 (height 3) taller than Child 4 (height 6)? No. Taller people in front:

$\boxed{0}$

For Child 5 (Orange, height 2): People in front: Child 1 (Yellow, height 1), Child 2 (Blue, height 4), Child 3 (Yellow, height 3), Child 4 (Green, height 6). Is Child 1 (height 1) taller than Child 5 (height 2)? No. Is Child 2 (height 4) taller than Child 5 (height 2)? Yes. Is Child 3 (height 3) taller than Child 5 (height 2)? Yes. Is Child 4 (height 6) taller than Child 5 (height 2)? Yes. Taller people in front:

$\boxed{3}$

For Child 6 (Red, height 7): People in front: Child 1 (Yellow, height 1), Child 2 (Blue, height 4), Child 3 (Yellow, height 3), Child 4 (Green, height 6), Child 5 (Orange, height 2). Is Child 1 (height 1) taller than Child 6 (height 7)? No. Is Child 2 (height 4) taller than Child 6 (height 7)? No. Is Child 3 (height 3) taller than Child 6 (height 7)? No. Is Child 4 (height 6) taller than Child 6 (height 7)? No. Is Child 5 (height 2) taller than Child 6 (height 7)? No. Taller people in front:

$\boxed{0}$

For Child 7 (Pink, height 5): People in front: Child 1 (Yellow, height 1), Child 2 (Blue, height 4), Child 3 (Yellow, height 3), Child 4 (Green, height 6), Child 5 (Orange, height 2), Child 6 (Red, height 7). Is Child 1 (height 1) taller than Child 7 (height 5)? No. Is Child 2 (height 4) taller than Child 7 (height 5)? No. Is Child 3 (height 3) taller than Child 7 (height 5)? No. Is Child 4 (height 6) taller than Child 7 (height 5)? Yes. Is Child 5 (height 2) taller than Child 7 (height 5)? No. Is Child 6 (height 7) taller than Child 7 (height 5)? Yes. Taller people in front:

$\boxed{2}$

Diagram 1

Step 3 — Second Arrangement: Verifying Numbers

For the second arrangement, the children are arranged from left to right as: Blue, Yellow, Red, Orange, Yellow-green, Purple, Green. The numbers they say are given as: 0, 1, 0, 2, 2, 2, 1. We will find a set of relative heights that matches these numbers. Let us assign ranks from 1 (shortest) to 7 (tallest) to the children.

Let $C_1$ be Blue, $C_2$ be Yellow, $C_3$ be Red, $C_4$ be Orange, $C_5$ be Yellow-green, $C_6$ be Purple, $C_7$ be Green. Let $N_i$ be the number child $C_i$ says.

$N_1$ (Blue) = 0: No one in front is taller than Blue. $N_2$ (Yellow) = 1: Blue is taller than Yellow. $N_3$ (Red) = 0: Red is taller than both Blue and Yellow. $N_4$ (Orange) = 2: Red and Blue are taller than Orange. Yellow is shorter than Orange. $N_5$ (Yellow-green) = 2: Red and Blue are taller than Yellow-green. Orange and Yellow are shorter than Yellow-green. $N_6$ (Purple) = 2: Red and Blue are taller than Purple. Yellow-green, Orange, and Yellow are shorter than Purple. $N_7$ (Green) = 1: Red is taller than Green. Green is taller than Blue, Purple, Yellow-green, Orange, and Yellow.

This gives the final height order from shortest to tallest: Yellow < Orange < Yellow-green < Purple < Blue < Green < Red.

Let us assign ranks based on this order: Yellow (C2): Rank 1 Orange (C4): Rank 2 Yellow-green (C5): Rank 3 Purple (C6): Rank 4 Blue (C1): Rank 5 Green (C7): Rank 6 Red (C3): Rank 7

Now, we verify the numbers using these ranks.

For Child 1 (Blue, Rank 5): People in front: None. Taller people in front: 0.

For Child 2 (Yellow, Rank 1): People in front: Blue (Rank 5). Blue (Rank 5) is taller than Yellow (Rank 1). Taller people in front: 1.

For Child 3 (Red, Rank 7): People in front: Blue (Rank 5), Yellow (Rank 1). No one is taller than Red (Rank 7). Taller people in front: 0.

For Child 4 (Orange, Rank 2): People in front: Blue (Rank 5), Yellow (Rank 1), Red (Rank 7). Blue (Rank 5) and Red (Rank 7) are taller than Orange (Rank 2). Taller people in front: 2.

For Child 5 (Yellow-green, Rank 3): People in front: Blue (Rank 5), Yellow (Rank 1), Red (Rank 7), Orange (Rank 2). Blue (Rank 5) and Red (Rank 7) are taller than Yellow-green (Rank 3). Taller people in front: 2.

For Child 6 (Purple, Rank 4): People in front: Blue (Rank 5), Yellow (Rank 1), Red (Rank 7), Orange (Rank 2), Yellow-green (Rank 3). Blue (Rank 5) and Red (Rank 7) are taller than Purple (Rank 4). Taller people in front: 2.

For Child 7 (Green, Rank 6): People in front: Blue (Rank 5), Yellow (Rank 1), Red (Rank 7), Orange (Rank 2), Yellow-green (Rank 3), Purple (Rank 4). Red (Rank 7) is taller than Green (Rank 6). Taller people in front: 1.

All the given numbers for the second arrangement are correct.

Answer

(i) The rule is: The number each child says represents the number of people standing in front of them who are taller than them. (ii) For the first arrangement (shown in the image), the numbers each child says are:

First child (yellow): 0

Second child (blue): 0

Third child (yellow): 1

Fourth child (green): 0

Fifth child (orange): 3

Sixth child (red): 0

Seventh child (pink): 2 (iii) For the second arrangement, the numbers each child should say are:

First child (blue): 0

Second child (yellow): 1

Third child (red): 0

Fourth child (orange): 2

Fifth child (yellow-green): 2

Sixth child (purple): 2

Seventh child (green): 1

Question 4

Answer

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