Question 10
A triangular arrangement of numbers is called a smart triangle of order $n$ if one can arrange all the positive integers from $1$ to $\frac{n(n+1)}2$ in a triangular structure such that, apart from the bottom row of $n$ numbers, each number in a subsequent row above it is the absolute difference or the sum of the two numbers immediately below it.
The triangular structures represented below, from left to right, are some examples of smart triangles of order 2, 3 and 4, respectively.
(a) Does there exist a smart triangle of order 5, with the numbers 1, 2 and 3 in the vertices, as shown in the diagram below? If your answer is yes, please give an example. If your answer is no, please prove it is impossible. (10 marks)
(b) Does there exist a smart triangle of order 6? If your answer is yes, please give an example. If your answer is no, please prove it is impossible. (30 marks)
Original response sheet (continued on the next source page):
(a) My answer is yes and the example is shown on the right. / My answer is no and the proof is:
(b) My answer is yes and the example is shown on the right. / My answer is no and the proof is:
Paper instruction: Give a full solution, subject to the example-or-proof instructions in the question.
The triangular structures represented below, from left to right, are some examples of smart triangles of order 2, 3 and 4, respectively.
(a) Does there exist a smart triangle of order 5, with the numbers 1, 2 and 3 in the vertices, as shown in the diagram below? If your answer is yes, please give an example. If your answer is no, please prove it is impossible. (10 marks)
(b) Does there exist a smart triangle of order 6? If your answer is yes, please give an example. If your answer is no, please prove it is impossible. (30 marks)
Original response sheet (continued on the next source page):
(a) My answer is yes and the example is shown on the right. / My answer is no and the proof is:
(b) My answer is yes and the example is shown on the right. / My answer is no and the proof is:
Paper instruction: Give a full solution, subject to the example-or-proof instructions in the question.