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Problem of the Week
Problem C and Solution
Everything in its Place 1

Problem

  1. A Venn diagram has two circles, labelled A and B.

    Each circle contains integers that satisfy the following criteria.

    The overlapping region in the middle contains integers that are in both A and B, and the region outside both circles contains integers that are neither in A nor B.

    In total this Venn diagram has four regions. Place integers in as many of the regions as you can. Is it possible to find an integer for each region?

  2. A Venn diagram has three circles, labelled A, B, and C.

    Each circle contains pairs of integers that satisfy the following criteria.

    In total this Venn diagram has eight regions. Place pairs of integers in as many of the regions as you can. Is it possible to find a pair of integers for each region?

Solution

  1. We have marked the four regions W, X, Y, and Z.

    Region W is inside both circle A and circle B. Region Y is inside B and outside A. Region W is inside A and outside B. Region Z is outside both A and B.

    We plot the given fractions on a number line as a reference:

    A number line with reference points negative 2, negative 1, and 0 from left to right. The number negative seven-sixths is plotted between  negative 2 and negative 1. The number negative one-quarter is plotted between negative 1 and 0.

  2. We have marked the eight regions S, T, U, V, W, X, Y, and Z.

    A Venn diagram with three overlapping circles labelled A, B, and C and the eight different regions marked as described in the list that follows.

    It is helpful if we first think about the pairs of integers that each circle could contain. Two integers have a negative sum if they are both negative, or they have different signs and the negative number is larger in magnitude than the positive number. Two integers have a negative product if they have different signs. Two integers have an even difference if they are both even or both odd, regardless of their signs, and regardless of which number is being subtracted from the other.