We Will Graph You! (“quadratic version”) words by Lawrence Mark Lesser; (can be sung to the tune We Will Rock You)
You wanna draw a parabola
From the general form algebra.
Don’t despair, complete the square:
The x of the vertex comes out there, and
(We will, we will graph you!)2x
The vertex now is figured out, / But does the graph smile or frown'
The number next to the square of x / Gave the sign to make the sketch, now
(We will, we will graph you!)2x / Find and plot any x-intercepts,
The constant “c” is the y-intercept; / A vertical line through the vertex
Gives symmetry for your sketch, and
(We have, we have graphed you!)2x
PUZZLE 1: A room is an equilateral triangle with each side measuring 100 meters. It is divided into 100 rooms, all equilateral triangles with sides of 10 meters. Each interior wall between two rooms has a door. If you start inside one of the rooms and can only pass through each door once, what’s the maximum number of rooms can you visit'
PUZZLE 2: Niko claims she has found three positive numbers whose product is one and their sum is greater than the sum of their inverses. She also claims that just one of the numbers is greater than 1. Is she right'
PUZZLE 3: Nine examiners each award 20 competitors a rank from 1 to 20. The competitors’ score is the sum of the ranks from the 9 examiners, and the winner is the competitor with the lowest score. For each competitor the difference between the highest and lowest ranking (from different examiners) is at most 3. What is the highest score the winner could have obtained'
Solutions on September 12
Sohini Majumder, Calcutta; Partha Sarathi Sen, Agarpara; Mrityunjay Ganguli, Burdwan; Shabnam Khatun, South 24-Parganas; Anupam Sengupta, Behala; Siddharth Khemka, Howrah; Bhujbal Sinha, Ranchi; Manu Sawant, New Delhi; Ajay K. Sharma, Burdwan; Purandar Khosla, Calcutta; Adhir Pratap, Calcutta; Phani Mishra, Bihar
Manisha Arora, Chandigarh; Chimay Mukherkee, Barasat; Anusuya Sanyal, Baruipur; Amitava Maity, Midnapore; Kajol Batabyal, Bankura; Rajdeep Dharitriputra, Orissa; Aritra Das, Howrah; Ajay K. Sharma, Burdwan; Manoj K. Khaitan, Calcutta; Amit Bakshi, Calcutta, Anupam Sengupta, Behala; Puran Ghosh, Taki
Please send your entries to firstname.lastname@example.org within 10 days. Snail mail address is Brainstorming, The Telegraph, 6, Prafulla Sarkar Street, Calcutta-700 001. Please send complete solutions, not one-line answers.
The response for puzzles posed on July 15 was lukewarm. We didn’t have too many entries.
Solution 1: 30.
Solution3: The bug crawls three edges.
Explanation for puzzle 1: The second player can play the following strategy: (1) if the first player plays 2n-1 for 1<=<=9, then he replies 2n with the opposite sign; (2) if the first player plays 2n for 1<=<=9, then he replies 2n-1 with the opposite sign; (3) if the first player plays 19 or 20, then he plays the other with the same sign. This secures a score of at least 39 [from (3)] less 9x1 [from (1) and (2)]. So he can ensure a score of at least 30.
Since the 1st player has a strategy to do no worse than 30 and the 2nd player has a strategy to do no worse than 30, these strategies must actually be optimal.