Reasoning · Grade 4-1 Arithmetic Puzzles
Problem
Name the cells and single out the centre.
Giving the cells names turns the picture into two short addition sentences that can be compared side by side.
4.OA.A.3Make A Systematic ListAdding both lines counts the centre twice.
Looking at the whole cross at once, instead of one line at a time, is what exposes the special role of the overlapping cell — it is the only number paying twice.
4.OA.A.3Change Focus Count The ComplementAdding the two lines together counts the centre cell twice, which is what singles it out.
The centre lies on both lines, so it is counted once from each of them while every other cell is counted only once.
The combined total is all five numbers plus one extra copy of the centre, so the extra copy can be peeled off to leave a plain sum.
So one line totals half of 15 plus the centre.
Splitting a total into two equal parts is only possible when the total is even, and odd-plus-odd-is-even is a fact you can see by pairing counters up.
2.OA.C.3Eliminate PossibilitiesTo halve exactly the centre must be odd.
Three requested answers and exactly three surviving centre numbers is a strong hint that the elimination has been done correctly.
2.OA.C.3Eliminate PossibilitiesCentres 1, 3, 5 give line totals 8, 9, 10.
Halving the combined total gives one line's total, because the two lines are equal by the rule of the puzzle.
4.OA.A.3Make A Systematic ListWith centre 1 the rest split as 2 + 5 and 3 + 4.
Subtracting the centre from the line total tells you exactly what the remaining pair must add to, so the search becomes a two-number sum you already know by heart.
2.OA.B.2Make A Systematic ListWith centre 3 they split as 1 + 5 and 2 + 4.
The same one-step method works for every centre value, which is what makes the three cases quick once the first one is understood.
2.OA.B.2Make A Systematic ListWith centre 5 they split as 1 + 4 and 2 + 3.
Each case ends with the four leftovers splitting perfectly into two equal-sum pairs, which is the signal that the arrangement really works.
2.OA.B.2Make A Systematic ListSwapping places repeats an answer, so there are 3 ways.
Knowing which rearrangements count as the same answer keeps you from thinking there are more than three solutions when there are not.
4.OA.A.3Make A Systematic ListThe cell where the two lines cross gets counted twice — spot that, and the whole puzzle shrinks to three easy cases!