Dr J Frost (jfrost@tiffin.kingston.sch.uk) KS3 Number Grids Dr J Frost (jfrost@tiffin.kingston.sch.uk) Last modified: 8th December 2014

Starter In the Maths Challenges and Olympiads, you are often asked to insert the numbers 1 to 𝑛 into a structure such that the sum of each row/column/line in the structure is the same. Q The numbers 1 to 9 all appear once in the following structure. The sum of each line is the same. What are the missing numbers? 1 3 5 6 4 9 ? ? ? ? 8 7 2 ? ?

General Tips Q The numbers 1 to 9 all appear once in the following structure. The sum of each line is the same. What are the missing numbers? Bro Tip 1: Calculate the sum of 1 to 𝑛. The quick way is 1 2 𝑛 𝑛+1 1 3 1 2 ×9×10=9×5=45 ? 7 Bro Tip 2: Let 𝑇 say be the total in each line. Form an equation, one side in terms of 𝑇 and the other side in terms of numbers used (considering overlap) ? 4𝑇=45+1+7+3 (if we add the 4 lines, 1, 7 and 3 are each seen twice) 4𝑇=56 𝑇=14 This then makes the problem much easier, as the middle two numbers are now obvious, and there’s little to experiment with in the outer boxes.

More Tips Q2 Again, the numbers 1 to 9 appear in each box. Let 𝑻 be the total of each row. Find the possible values of 𝑻 Bro Tip 3: Give any overlapping squares a (separate) variable name. 1 3 ?  See diagram. Form an equation, one side in terms of 𝑇 and the other side in terms of numbers used (considering overlap) 𝑥 ? 4𝑇=45+1+3+𝑥 4𝑇=49+𝑥 Bro Tip 4: Reason about divisibility on each side of the equation. ? LHS is divisible by 4, so RHS must be. Thus 𝑥=3, 7. But 3 already used, so 𝑥=7, and thus 𝑇=14

Final Tip Q3 Again, the numbers 1 to 9 appear in each box. Let 𝑻 be the total of each row. Find the possible values of 𝑻 Give any overlapping squares a (separate) variable name. 𝑦 𝑥 ?  See diagram. 1 Form an equation, one side in terms of 𝑇 and the other side in terms of numbers used (considering overlap) Bro Tip 5: Consider the smallest and largest your variables could be to give a plausible range for 𝑇. 4𝑇=45+1+𝑥+𝑦 4𝑇=46+𝑥+𝑦 ? What would we do at this point? The smallest 𝑥 and 𝑦 could be is 2 and 3 (1 is used) 4𝑇=45+1+2+3=51 → 𝑇=12.75 So 𝑇 is at least 13. The largest they could be is 8 and 9. 4𝑇=45+1+8+9=63 → 𝑇=15.75 So 𝑇 is at most 15. i.e. 13≤𝑇≤15 ? ? We’ve only shown 𝑇 can plausibly be between 13 and 15, but we haven’t shown these values actually work. Consider each value of 𝑇 in turn and show it can lead to a valid arrangement.

Olympiad Worksheet Worksheet provided. Here is a reminder of the tips: Full solutions on next slides. Bro Tip 1: Calculate the sum of 1 to 𝑛. The quick way is 1 2 𝑛 𝑛+1 Bro Tip 3: Give any overlapping squares a (separate) variable name. Bro Tip 2: Let 𝑇 say be the total in each line. Form an equation, one side in terms of 𝑇 and the other side in terms of numbers used (considering overlap) Bro Tip 4: Reason about divisibility on each side of the equation. Bro Tip 5: Consider the smallest and largest your variables could be to give a plausible range for 𝑇.

J25 ?

J38 ?

J24 ?

J48 ?

M30 ?

M48 ?

M66 ?

M72 ?

M65 ?