Holiday meeting point: minimize total distance on a line

Holiday meeting point choices can be explored with a small fictional line model. Put five imagined starting positions at coordinates one, four, seven, nine and fifteen. Choose a real coordinate x for a shared meeting point. The objective is to minimize the sum of the five absolute distances to x. Under these exact rules, coordinate seven is the unique best location, with total distance nineteen.

This is an original mathematical exercise for a holiday planning worksheet. The coordinates describe no actual neighborhood, journeys or people. There are no roads, obstacles, journey times, unequal costs or restrictions on which real coordinate may be selected. Those details would change an actual planning decision. Here the declared distance objective is enough to prove a precise answer.

Write the distance objective before choosing a location

For a starting coordinate a, its distance to x is the absolute value of x minus a. Absolute value makes the distance nonnegative regardless of direction. For example, the distance between one and seven is six, and the distance between fifteen and seven is eight. Do not add a signed negative distance for a point on the other side.

Add one distance for each of the five entries. At x equal to seven, the distances in input order are six, three, zero, two and eight. Their sum is nineteen. That calculation proves nineteen can be achieved. It does not by itself prove every other location costs more; the lower-bound argument below supplies that missing part.

All five entries have equal weight. A coordinate records one starting position in this fictional model, rather than a group size. The objective adds distances, with no squaring or maximum-distance rule. Keeping that sentence beside the calculation prevents a different notion of a good meeting point from being silently substituted.

Pair the two outside coordinates

Begin with the contributions from one and fifteen. Wherever x is chosen, its distances to those two points add to at least fourteen, the distance between the endpoints. If x lies between one and fifteen, the two distances partition that fourteen-unit span. Their sum is exactly fourteen throughout the closed interval from one to fifteen.

If x is less than one, both endpoints are to its right. The sum becomes one minus x plus fifteen minus x, or sixteen minus twice x. That is greater than fourteen when x is below one. Moving outside the pair has added an extra outward distance that appears in both contributions.

If x is greater than fifteen, the distances become x minus one and x minus fifteen. Their sum is twice x minus sixteen, again greater than fourteen. Equality for this pair therefore occurs exactly when x is at or between its endpoints. The inclusion of both endpoints matters when checking equality conditions.

The same reasoning applies to any pair a at most b: the sum of the distances from x to a and b is at least b minus a. Equality holds precisely for x in the closed interval from a to b. We will use both the bound and its equality interval, rather than keeping only the numerical total.

Pair the next two and retain the middle contribution

The next pair is four and nine. Its contribution is at least five, with equality exactly when x lies between four and nine, inclusive. These points were not selected by guessing a likely meeting location. They are the remaining outermost entries after pairing one with fifteen.

Only coordinate seven remains unpaired. Its contribution is the absolute value of x minus seven. Combining the two pair bounds gives total distance at least fourteen plus five plus that remaining absolute value. In other words, the total is at least nineteen plus the distance from x to seven.

At x equal to seven, both pairs meet their equality conditions. Seven lies inside the interval from one to fifteen and inside the interval from four to nine. The leftover distance to seven is zero. Thus the lower bound nineteen is attained by the location already checked directly.

For every x different from seven, the leftover distance is positive. The total must then exceed nineteen, even if both paired contributions remain at their lower bounds. This establishes uniqueness. It rules out every other real coordinate, including fractional positions, rather than checking only the five starting coordinates.

Compare a nearby location without confusing the objective

At x equal to six, the distances are five, two, one, three and nine. Their sum is twenty. Both pairs still contribute their minimum totals, fourteen and five, while the unpaired seven contributes one. This provides a direct check of the bound near the optimum.

At x equal to eight, the distances are seven, four, one, one and seven. The sum is again twenty. Equal totals at those two locations do not make every location between them equally good. The leftover distance reaches zero only at seven, where the total falls to nineteen.

The arithmetic mean of the five coordinates is thirty-six divided by five, or seven point two. At that location the distances are six point two, three point two, zero point two, one point eight and seven point eight. They add to nineteen point two. It is a valid candidate with a larger total than nineteen.

This comparison concerns the stated sum of absolute distances. It supplies no rule that an arithmetic mean is generally useless or that one statistic answers every planning question. A different objective would need its own proof. Keep the exact objective with the coordinate chosen from it.

Remove the middle entry and identify the full answer interval

Now define a separate four-entry exercise using one, four, nine and fifteen. Coordinate seven is no longer an input. Pair one with fifteen and four with nine as before. Their contributions still have lower bounds fourteen and five, so the total is at least nineteen.

There is now no unpaired term forcing x to a single location. Both pair bounds are attained whenever x belongs to both equality intervals. The intersection of the intervals from one to fifteen and from four to nine is the entire closed interval from four to nine.

At x equal to four, the four distances are three, zero, five and eleven, totaling nineteen. At x equal to nine, they are eight, five, zero and six, also totaling nineteen. At x equal to six point five, the distances are five point five, two point five, two point five and eight point five, again nineteen.

Every real location from four through nine is optimal in this separate exercise. Selecting the midpoint six point five gives one valid optimum, but it does not establish a unique answer. If a worksheet requests all minimizers, retain the interval rather than replacing it with one convenient representative.

A location below four makes the inner pair's contribution exceed five. A location above nine does the same on the other side. The outer pair cannot contribute less than fourteen to compensate. Thus locations outside the stated interval have total greater than nineteen, completing the description of all minimizers.

Generalize the pairing while preserving the assumptions

For any sorted odd number of real input coordinates, pair the smallest with the largest, then continue inward. Each pair contributes at least its endpoint separation. The middle coordinate remains alone. That middle coordinate lies inside every paired interval, so choosing it attains every pair bound and makes the leftover distance zero.

Any other choice makes the leftover distance positive. Therefore the middle ordered coordinate is the unique minimizer of the equal-weight sum of absolute distances for an odd number of entries. Repeated coordinates remain separate entries; the same argument still applies when some paired endpoints coincide.

For a sorted even number of entries, pairing leaves no single middle term. Every pair reaches its minimum throughout its endpoint interval. The intersection of those nested intervals runs from the lower middle coordinate to the upper middle coordinate. That complete closed interval is the set of minimizers.

If the two middle coordinates coincide, their interval contains one point. An even number of entries therefore does not guarantee multiple different optimal coordinates. The result depends on the actual two middle values. Preserve duplicates and their positions while sorting so this case is handled accurately.

A finite control set also checks the worksheet arithmetic. It includes every sorted multiset of four or five integer coordinates chosen from zero through six, allowing repeats. For each of these 672 inputs, checking half-unit locations from negative one through seven finds exactly the expected median point or middle interval on that grid. The pairwise proof supplies the conclusion for all real locations, beyond that finite check.

For a concrete repeated-entry check, take the separate four-coordinate list two, two, two and five. Its two middle coordinates are both two, so the minimizing interval collapses to that single coordinate. At x equal to two, the distances are zero, zero, zero and three, totaling three. At x equal to three, they become one, one, one and two, totaling five. Choosing an arbitrary point between the smallest and largest would therefore fail.

For an odd repeated-entry example, take two, two, five, five and five. The middle ordered entry is five. Pairing the extremes gives a three-unit bound twice, with a final zero contribution at five, so the minimum is six. At x equal to two the total is nine. Repetition changes how many contributions arise at a location, even though it does not create a new distinct coordinate. Counting only distinct values would change the objective by discarding entries.

Keep the neighboring exercises within their own questions

The median and mean puzzle-time comparison reports an arithmetic average and ordered middle from six fictional completion times. It addresses what a descriptive summary means. This worksheet instead proves which freely chosen coordinate minimizes an explicitly defined absolute-distance total.

The minimum step cost exercise reaches an exact integer total using allowed moves and compares last-move predecessors. Its recurrence does not choose a real meeting location. Here the proof comes from pair bounds and their equality intervals, with no sequence of allowed moves.

Read the two native towel accounts with their limits

The plum six-piece towel listing describes two bath towels, two hand towels and two washcloths. All six images show plum textiles, including different sizes, a folded stack and marketing demonstrations. Bathroom objects are context props. Cotton, GSM, dimensions, absorbency and durability statements remain unverified listing claims.

The white thirty-six-pack towel listing has five images of white textiles, folded stacks and lifestyle uses. The bulk count is a listing statement, not an independently checked delivery. Material blend, measurements, weight, softness, drying and care performance are unverified. People, pet and equipment shown are not established supplied contents.

Keep an unrelated transaction enquiry separate

A reader considering an unrelated purchase on republeq.com could ask whether USDT is offered as a cryptocurrency payment method for the intended order. Such an enquiry would need to identify the exact offer and quantity so any answer concerns that transaction. The coordinates in this exercise are invented mathematical inputs; they are neither prices nor token amounts. Before deciding whether to proceed, the reader could review the actual choices and terms presented for the proposed purchase and ask the responsible merchant to clarify essential missing information. If a cryptocurrency option is available, the transaction information would need to specify the asset, network, receiving destination, amount and reference linking a payment to the order. A token name alone would not settle those details. Any delivery, cancellation, return or refund question would also need an answer for that same offer, including the applicable refund denomination and conditions. An answer about an earlier basket would not establish the terms of a changed order. This article gives no wallet address, transfer instruction, conversion rate, fee estimate or assurance of recovery. It promises no current USDT acceptance or transaction outcome. The reader can finish the mathematical worksheet without making a purchase. If an order or payment status is uncertain, clarify that status through the responsible merchant before considering another payment. Keep any eventual acknowledgement and transaction record with its own order information. A correct distance minimum cannot establish that funds were sent, accepted or reconciled, and a payment record cannot validate a proposed location. Each question needs the evidence appropriate to the decision being made.

For the original five-entry holiday meeting point exercise, save coordinate seven and total nineteen with the pair bounds that prove them. For the separate four-entry version, save the full interval from four to nine. The difference comes from the unpaired middle contribution, which forces one location in the odd case and is absent in the even case.